in what form did William the Conqueror write the date of his own coronation?

In general, dating was complicated, and different conventions existed simultaneously in England at that time.

For the specific example of William the Conqueror's coronation, we have different sources within the following decades implying that it was in 1066 or 1067, anno Domini. The precision sought in the question did not exist, at least in the same form that we think of modern dates as precise. The way of recording the date and the year was simply different from our modern convention of December 25th, 106X.

The simple answer is that we don't know how William would have written his own date, because all we have are chroniclers writing about the event after the fact. We don't have a lot of documents from his reign, so we can only speculate about the way the seven days mentioned here would have been referenced. However, the problem of years with various lengths and different calendars coexisting (not to mention multiple occurrences of the same date within one year) were familiar ones to English chroniclers of the time, so we can say how they would deal with these dates with a good deal of certainty.

For the long answer, it's really, really, really complicated. I'll try to summarize the basic issues here, since anyone working with medieval dates in Europe should be aware of these potential problems.

A summary of date recording practices in England can be found here. This is only the first chapter and introduction for a longer book; for details, refer to the rest of this book.

For many dated documents, the year might not even be given, since it could be assumed from context. When a document did want to be very precise, it would often give a litany of dating methods to establish the year. A medieval English diploma which might give a formal date could list a number of possible ways to establish the year, and usually contained a number of these:

1. The indiction year. This gives the year number in a cycle of 15 years, according to an ancient Roman method. There were a number of different possible indiction year cycles which were assumed in various places, but this gives you a sense of the relative year if you know which cycle is being referenced. Historically they tended to begin at some date in the fall (generally September 1st or September 24th), and this practice was still found in isolated cases in England. The Vatican adopted a standard starting date of January 1st in the medieval period.
2. The regnal year of the monarch. Regnal years could be counted from the day a king ascended to throne, or they could be counted from the day of the coronation. When the year began could depend on the monarch's choice about when his year began (or sometimes on how the local scribe decided to count). Regnal years are sometimes particularly complicated when they begin on a movable feast. For example, King John's regnal year began on Ascension Day, which is tied to the date of Easter, which obviously moves from year to year. Thus, the lengths of John's regnal years would vary by up to a month or so, and many of those years had multiples of the same calendar date contained within them.
3. The regnal year of the pope.
4. The regnal year of some other random person (local aristocrat, etc.).
5. The golden number. Each year is given a number within a 19-year ("Metonic") cycle that has to do with computing the date of Easter (and the lining up of solar years with lunar cycles).
6. The epact. A number which describes the phase of the moon on March 22nd of a given year, which is also used to compute Easter and has a relationship to the golden number.
7. The dominical letter. A letter from A to G used to designate the day of the week that January 1st fell on. (The use of January 1st as the standard here is pretty consistent, following the Vatican indiction cycle mentioned above.) Note that the dominical letter would change in mid-year during a leap year on February 24th, so leap years get two dominical letters.
8. The concurrents. The number of days (1 to 7) between the last Sunday of the preceding year and new year's day (which here was often taken to be March 25th). This has an obvious relationship to the dominical letter.
9. Some other random year description relating to a well-known event.

Oh, yes, and a year could also be dated by:

• The year given as anno Domini ("In the year of [our] Lord") or equivalently, anno gratia ("In the year of [God's] grace").

The description anno Domini was not a standardized concept and doesn't necessarily show up in English dates that often in this period. Most of the other dating methods were actually more helpful in relating events to useful other things going on or to short cycles of years, rather than measuring against some event that occurred a millennium before.

When anno Domini does show up, the year could begin on:

• January 1st - often used for legal purposes
• December 25th - the day Jesus was born, therefore a logical place to begin counting "years of our Lord"
• March 25th of the year preceding January 1st - the day Jesus was conceived (the Feast of the Annunciation), the logical dating if we assume the calendar is supposed to begin when Jesus's life supposedly began in the womb, used in a minority of places
• March 25th of the year after January 1st - an illogical move of the calendar forward to the Annunciation rather than back, this was nevertheless the more common implication in England at this time when March 25th was used for the new year
• Martinmas (the Feast of St. Martin, November 11th) or Michaelmas (the Feast of Michael the Archangel, September 29th) - both of these dates were often used as the beginning of the year for legal or financial systems, based on old Germanic traditions, which survived in some parts of the English government even after the Norman Conquest and in rural areas
• Easter Day - which is complicated because it moves from year to year (more common on the Continent than in England)
• There are less common options, like September 1st or 24th, sometimes used to begin Indiction years, March 1st in old Saxon tradition sources, etc.

Then, once you've established the year, you would need to give a day within that year. There are various methods of representation and problems that could crop up here, too:

• Dates were often given in relationship to some church feast. This is fine for dates like "The Nativity of Christ" (i.e., Christmas) which equals December 25th, presumably in some particular calendar year. But when you date something to "the fourth feria after the second Sunday before Easter," that date could vary from year to year. (A "feria" refers to the days numbered after Sunday when a major church feast did not occur, so the "fourth feria" would be Thursday.) If the year begins on March 25th, you could also have a situation where that specific liturgical date actually occurs twice within the same calendar year.
• Our modern way of stating a date by month and then day within that month (e.g., April 10th, August 2nd) was very uncommon in this period, but is sometimes seen.
• The ancient Roman method of dates was commonly followed in formal dating, where dates were counted backward from the important Roman dates within the month: the Calends (1st of the month), Nones (5th or 7th, depending on the month), and Ides (the 13th or 15th). Thus, December 25th would be recorded formally as "the 8th day before the Calends of January." Note that this follows the Roman convention of counting inclusively, where you include both January 1st and December 25th in enumerating the days to get up to 8.
• Although I don't remember seeing specific evidence of this, I'm pretty sure you might end up with some confusion in year dating for days in late December. Why? Because if we assume, for example, that the new year falls on January 1st, do we refer to the date as the "8th day before the Calends of January in 1067" or as the "8th day of 1066 before the Calends of January [in 1067]"? When anno Domini years get mixed up with ancient Roman dating systems, some weird things can happen.
• Oh, and just for extra credit for the Roman calendar, leap year wasn't on February 29th in the ancient Roman calendar. Instead, you doubled the "6th day before the Calends of March," i.e., February 24th. In leap years, there were two "6th days before the Calends of March," which has led to a number of dating translation errors when modern scholars read dates toward the end of February.

Now that we've reviewed the various potential ways to record a date in medieval England, we can address the specific questions.

One potential problem is that the question assumes the existence of a standard calendar operating at least throughout England. This is not a good assumption for the 11th or 12th centuries.

In the question it is asserted that William moved the new year to January 1st. I don't know what evidence there is of this. (I'm not an expert on this period, though I have experience working with medieval sources.) I've seen this claim made in a number of 19th century calendar books and general knowledge "encyclopedias," and it is still stated in less scholarly modern books. But we have clear examples of dates from the period that are inconsistent within England, so if William did make a declaration to change the calendar, it probably only affected some circumscribed area of the government. Local churches and monasteries, and perhaps even local governments, might continue to use whatever local form of dating they always had.

From the link above:

The reckoning from Christmas was soon in general vogue [following Bede]... The Anglo-Saxon and Norman kings of England used it after it had been abandoned in most quarters of Plantagenet England; as late as even the fourteenth century, the Chronicon de Lanercost still used the Nativity style.

So it appears that William's edict, if it existed, didn't have widespread immediate adoption.

For those scribes who actually may have experienced this new year's date shift (if it happened), they would likely just add additional year descriptions if the date was not clear from context. For example, "December 27th in the first year of the reign of our King William" is pretty clear in being two days after his coronation, regardless of the anno Domini year. In most cases, anno Domini years wouldn't even be that relevant.

Similarly, I can't seem to find particular documentation that everyone in England suddenly changed the dates of the new year in 1155. This date seems arbitrarily to line up with Henry II becoming the first Plantagenet king of England. Again, this claim often shows up in encyclopedias and such in the 19th century, giving a false sense of standardization. There may have been some shift or impetus to change the calendar in some areas of the government, but again this wouldn't necessarily affect local dating customs throughout England.

From the link above:

In England [the March 25th new year] is found as early as the middle of the eleventh century, when certain annals of the Anglo-Saxon chronicle were apparently dated by this reckoning, but it only came into common use in the twelfth century and so continued until 1752.

In general, when the March 25th date was used in England, it refers to a delayed start to the new year, rather than the logical reference to nine months before (though apparently isolated cases of both practices have been found from this period in England).

In sum, the Nativity (December 25th), January 1st, and the Annunciation (March 25th) were all used during this period as dates for the beginning of the year. The practices overlapped historically and could vary from region to region, or even from church to town hall in the same place in England.

I sense that perhaps the concern in the question comes from a discomfort with the idea that a year could vary in length, but this was accepted as commonplace, for example in year dates -- like the regnal years of King John mentioned above -- that began on movable feasts.

Scribes from this era could be very precise about the dates if they wanted to be. But dating things to anno Domini wasn't always the priority. Even if there was a sudden date shift somewhere in these specific instances, scribes could just clarify the year with other references.