Does airport immigration ask for boarding pass on arrival?

Generally only if it's relevant for obtaining a visa on arrival, such as in the case of US citizens flying into Baku on the direct AZAL flight from New York.

Another case is if flying into Ireland from the UK, in which case a boarding pass proving you flew from the UK will exempt you from having to present a passport (if you're believably British or Irish).

On departure, however, many countries systematically require it, and some, such as Moldova and Georgia, stamp it.

Use your Chinese passport throughout the journey. Sri Lanka can report to China on your dual status for using an invalid Chinese passport to enter Sri Lanka because according to article 9 of the nationality law you have already cease to become a Chinese citizen with PRC upon naturalising as a UK citizen. You are effectively illegally entering Sri Lanka and China on an invalid passport.

I have only ever been asked to show my boarding pass on arrival once. It was on a flight from Ireland to the UK.

The reason was that Ireland is the only country you can enter the UK from without going through passport control (due to the border between the Republic of Ireland and Northern Ireland), and this was a simple way to verify that passengers had come from Ireland.*

(Despite thisn exemption, I suspect any passenger acting suspiciously would be taken to one side for questioning.)

The above obviously would not apply in the OP's case. I would be surprised if it was asked for, as many people lose them as soon as they have found their seat. But you never know.

*Schengen area airports generally physically separate passengers arriving from the Schengen area and those arriving from other countries who do have to go through passport control. As the UK and Ireland are not part of the Schengen area but instead have their own separate agreement, the volume of international passengers not required to go through passport control is so small that they do sometimes use this ad hoc method of checking boarding passes.

The boarding pass would only be an issue if it were a requirement for the type of visa you would need to enter Sri Lanka (personally I have never faced such a requirement).

However, even if you wanted a transit visa - the officer would be asking for your flight itinerary/reservation rather than boarding passes.

On arrival to Sri Lanka I want to show and have the entry stamp in my UK passport so when I travel onward to London a few days later there is no exit formality confusion.

On arrival in Sri Lanka you only need to show that you have the proper visa (or are exempt from such requirements) to enter Sri Lanka. That's it.

So when I show my UK passport to SL immigration on arrival, I want to know if they check in a computer to see if that name matches the flight records (or more basically actually look at the boarding pass) to check?

Generally speaking immigration officials can ask for anything (I was once asked what color are my eyes - I still don't know why); but they will definitely not check that your name matches the boarding pass.

They will check:

1. Photo matches the person standing in front of them.
2. Passport passes the machine reader.
3. You are not blacklisted or otherwise forbidden to enter Sri Lanka
4. Your passport is valid.

Then they will ask (these are the standard questions, but as I mentioned they can ask anything):

5. Where are you coming from.
6. Purpose of your visit.
7. Where are you staying.
8. Itinerary (especially if you are asking for a transit/short stay visa)
9. Proof of funds (optional, but sometimes asked on certain visa types).

However, on exiting the country if there are exit immigration requirements I know some Asian countries will stamp both the boarding pass and the passport with the exit stamp.

The officer will normally scan the boarding pass at this stage or manually inspect its details.

So, during your exit from Sri Lanka - you will need to show (at the airline counter) that you have valid travel documents for the UK (you can show your UK passport), and then further at immigration control a passport to mark your exit.

They may ask when you arrived in the country (especially if when they scan your UK passport they do not see a recent arrival record). At that time, you can show your Chinese passport as that is on which the arrival was recorded.

To avoid all this hassle - you should just use your Chinese passport throughout the journey; and then once you land in UK just show your UK passport at the immigration counter to speed your entry.

Sri Lanka does not know or care that you have multiple nationalities, so using your UK passport will be fine.

Immigration can ask pretty much anything they want to, but I can't recall ever having to show my boarding pass on arrival. If they did, and they noticed the name is different, there is nothing wrong with showing your Chinese passport as well.

That said, because China does not tolerate dual nationality, per the "Case 4" recommendation in this answer I would advise you to cover your tracks: use your Chinese passport to enter and leave Sri Lanka, and only switch to the UK passport when arriving in London. Chinese nationals are eligible for the same e-visa as UK nationals.