No, not required. 🙂 Just make sure that you have the proofs - even though they most likely will not ask to see it.

A UK-passport holder doesn't require a visa for a stay of up to 30 days.

At entry, the immigration officer can ask for proof of onward travel, booked accommodation and proof of finances (20,000 THB in cash or equivalent).

There are sometimes problems, when someone enters on a visa, gets an extension of stay, then leaves WITHOUT cancelling that and after that applies for a new visa abroad and wants to get a new extension of stay inside of Thailand.

Extension of stay is what you call it when you get an extension inside of Thailand (based on work, retirement, etc.). Because the original visa "ends", but the extension of stay allows you to stay in Thailand.

What visa were you on before you left?

I'm just guessing here, but could the actual problem be that you left the country without cancelling your previous extension of stay? Or did you do that?

You could also apply for a 30 day extension (1900 Baht) on your tourist visa and then go and apply for the ED visa in January. That way you can avoid travelling abroad during new year and you don't "waste" days on your tourist visa by leaving early.

Yes, up to 15 days before and 7 days after due date.

It depends on the school if they provide the necessary paperwork and the immigration office if they let you switch the visa type inside the country. You will always have the option to visit one of the many embassies/consulate around Thailand and get the ED visa there. The school will be able to provide you with that info. They usually know best, what they can do and what not.

If I was you, I would go and get a visa in Holland if you already plan to stay more than 30 days and you still have enough time to apply. It only costs around 1000 Baht there. The extension is 1900 Baht plus you will need time during your holidays. Also you can get an extension on top of your visa which allows you to stay up to 90 days in Thailand.

If you don’t want to do that: you can get an extension at any Immigration office (not at the airport). Do it after 15 or 20 days. Some won’t let you get the extension too early. You won’t lose any days. They will add the extension on top of your original expiry date.

You can only get one extension per stay, after that you need to leave the country.

If you got the papers already, you should see,

to what embassy/consulate those are addressed

Yep, usually, at least in Bangkok, the immigration will do your first 90 days report together with the extension of stay. Check for that paper that Robert posted.