Employer agreed notice period and now wants to extend it - UK

Once they all agreed, it was sent to HR and HR then sent a 'Confirmation of Resignation' email, confirming my final day as 3 weeks' time.

So long as you have a copy of that email, I think the company has no leg to stand on and your last day will be in 3 weeks time.

The main aspect is that you negotiated a finish date, in good faith and then would have made arrangements with your new employer, based on that good faith negotiation.

I would simply reply to the higher ups with a copy of the email - advise that you agreed an end date in good faith (this is important, because you are invoking legal language) and have advised your new company your start date based on that information and have signed an offer, with that start date.

If they threaten legal action, let them know that you consider their actions Tortious Interference and will be more than happy to counter-sue.

Is it reasonable for my employer to now expect me to extend my notice period after receiving explicit confirmation from my manager, their manager, and HR?

Absolutely not. They agreed in writing to a 3 weeks notice period. This supersedes the 3 months period from your employment contract or handbook. They can ask for a longer notice period, but you have every right to say no.

Would they be able to enforce the 3 month notice period if a 3 week notice period had already been explicitly agreed in writing by multiple parties?

No. Firstly a notice period cannot be enforced in a sense that some burly guys show up at your front door and forcibly drag you to work. The only option that the employer has is to sue you for breach of contract and/or damages. Since you have their written agreement, you are not in breach of contract and any damages are their problem since they agreed. In fact, THEY are in breach of contract and since you already have lined up a new start you could potentially sue them for damages if they insist.

Anyway, you are safe here. If they sue, they would get laughed out of court.

If it has an effect on you (like you signed to start with a new company in four weeks time), see an employment lawyer.

I assume that once your notice was accepted it was accepted, your employment lawyer will tell you definitely and will be able do send a most convincing letter for you. The fact that it is an employment lawyer and not little you is often enough. And your manager calling your behaviour rude frankly has some problems and look for treatment.