Can I interrupt quarantine to go get the 2nd dose of the vaccine after coming back from an amber list-country?

Firstly, referring to guidance (as linked to in other answers) can certainly be useful but you need to be careful as guidance does not generally carry the weight of law and often uses paraphrasing which can lead to subtle inaccuracies. If you want to be sure it’s always best to refer directly to the legislation.

Secondly, there is no such thing as a "UK" jurisdiction when it comes to law, so the exact answer depends on where you have arrived from abroad. The UK is made up of four jurisdictions: England, Wales, Northern Ireland, and Scotland. Any given law could apply to any combination of those.

The relevant law here for England is the Health Protection (Coronavirus, International Travel and Operator Liability) (England) Regulations 2021.

Regulation 9 provides that when you arrive in England from a so-called "amber" country (a "category 2" country in the regulations) you must usually self-isolate. Some people are exempt but I’ll assume that is not applicable here.

Regulation 9(11) provides the list of exceptions to the requirement to self-isolate. I won’t copy the entire list here (you can click the link above to see it), but suffice to say there is no specific exception for obtaining a 2nd dose of a vaccine. The following medical exception is available, but it’s hard too see that a 2nd dose could be considered urgent or that you will be able to get a doctor’s advice that you cannot wait until the self-isolation period has ended:

Regulation 9(11)(b): "to seek medical assistance, where this is required urgently or on the advice of a registered medical practitioner, including to access services from dentists, opticians, audiologists, chiropodists, chiropractors, osteopaths and other medical or health practitioners, including services relating to mental health"

There is also a catch-all "exceptional circumstances" provision at regulation 9(11)(i). This means that you can break your self-isolation for any reason (including one which isn’t specifically mentioned) provided that it constitutes exceptional circumstances. Whether it does or not would be a matter for the courts to decide if it came to that. The regulation gives some (non-exhaustive) examples of what might constitute exceptional circumstances. One of them is:

"to obtain basic necessities such as food and medical supplies for those in the same household (including any pets or animals in the household) where it is not possible to obtain these provisions in any other manner"

The problem here is similar to the first exception: it is hard to imagine that a 2nd dose is exceptional circumstances or even basic "necessities" since there is no compelling reason you can’t just wait a few extra days to get it.

As a side note, the above is a good example of where guidance can be inaccurate. From this answer you can see that the guidance says "get basic urgent necessities like food and medicines where you cannot arrange for these to be delivered", but as you can see the actual rule is wider than that – you need to be unable to obtain the necessity in any other manner, not just from a delivery.

No. When I went to get vaccinated recently in the UK I was explicitly asked if there was any reason I should be self-isolating, presumably with the result that if I had said "yes" I would have been turned away (or worse!).

Given the fact that there’s a large time window for you getting your second dose I don’t see that there’s any necessity to get it during your quarantine.

To me, that seems to be correct. But for certainty, you should call the medical centre where you wish to be vaccinated.

The quarantine procedure is set out in more detail on this page. The most relevant sections seem to be:

Going out
Unless you’re at risk of harm, you cannot leave the premises where you’re in quarantine. You must only exercise inside the place where you’re quarantining or in the garden. You cannot leave the place where you’re quarantining to walk your dog. You will need to ask friends or relatives to help you with this.

and:

Exceptional circumstances
You can leave your accommodation in certain exceptional circumstances. This includes needing to:

• get basic urgent necessities like food and medicines where you cannot arrange for these to be delivered
• travel to a COVID-19 testing site
• access critical public services including social services and services provided to victims (such as victims of crime)
• move to a different place for quarantine where you can no longer remain where you are

There may be other exceptional circumstances that allow you to leave your place of quarantine.

You could get advice from a medical or other professional to help decide whether your circumstances are exceptional and require you to leave your place of quarantine.

Even if you have an exception, you must continue to follow the general restrictions that apply.

In short: it is only permitted to leave your place of quarantine in "exceptional circumstances". There’s (unsurprisingly) no exhaustive list of what qualifies as "exceptional circumstances", but several examples are given. The given examples (it’s impossible or unsafe to remain where you are, you need food/medicine and it can’t be delivered, accessing services for victims of crime, complying with legal obligations) all seem to fall broadly into the category of "emergency".

There is a mention of getting medicine which can’t be delivered at home, but I don’t believe that exception would be applicable to a COVID vaccination. The reason being that the exception is to

get basic urgent necessities like food and medicines where you cannot arrange for these to be delivered

(emphasis added). The phrase “basic urgent necessities” leads me to believe this medicine exception applies specifically to obtaining medicine somebody needs (for example, to manage an existing condition). While I fully encourage everybody to get vaccinated as soon as they can, I find it extremely difficult to imagine a covid vaccination would be considered an “urgent necessity” for anybody outside of specific high risk groups.

However, as I said at the start of the answer (and as suggested in the guidance), you can call the medical centre responsible for your vaccination and take advice from a medical professional. I very much expect they’ll advise you to arrange to receive the second dose after your quarantine period ends, but I may be wrong.