Caveat: don't use the following description as a recipe to put your life abroad. My survival and citizen skills are equal to zero. As a developer I spend my time in the internet and don't know anything else. So I might broke some laws.
First thing first you better have a job arranged before you came here. I've spent CVs to many companies and none of them replied. Well, one did, they wasn't interested. I went to a pub-meetup and was asking about open positions when found my current employer. During my presence here I managed to visit the office as well and have a dinner with the team. Without that I wouldn't have much chance to get a job. Some says you should have a UK number at least to put it on the CV. That's a nice trick as well. When you know you'll have a salary here it's safe to say - you're almost here. Why? Unless you have a monthly income you can easily empty your resources and find yourself with an empty bank account.
As a Hungarian citizen I didn't had to do any paperworks before my move. There is nothing important you have to arrange that time. It comes later - I'll talk about it in a minute. Couple of weeks before you arrive to the island you should look for an apartment. I've tried many sites and ended up on http://www.spareroom.co.uk/. Here you can create an advertisement for yourself - defined by your requirements and attributes. And since it's a big city landlords will contact you - you just have to browse in the offers. In general all requires from you to visit the flat before you take it. However I asked to make Skype calls - fortunately many was happy to call me and it's a nice experience to get to know your landlords and roommates. I started looking for empty apartments but then changed my mind and searched for room sharing. It's cheaper and you'll have a nice company.
Job done, flat done - travel! Getting a flight ticket is just 4 mintes and you can get anytime. The best to book is when you agreed with your future landlord when you could move in to your room.
Here the very first thing you have to do is to ask for an NI (National Insurance) number. I called the office and got an appointment for the following week.
At this point you have your employee contract and tenancy contract - so your employer can write you a statement where they prove your local address. This is eligible to apply for a bank account. Go to the first office and ask for a basic account. It's enough to do online payment and general everyday things. It's not a credit card but you don't need one hopefully. The card will arrive in a week.
In the meantime you probably managed to done your NI application interview. That should arrive in 2-3 weeks. When it's in your hand you can opt-out from the Hungarian healthcare system and get a local EU Health Insurance card. For the Hungarian bureaucracy it's the best to create an online account.
And that's all. If you count it's not more than 6 administration steps. It's a bit surprising how easy to move one's life to a new place. So huge thanks for being part of the European Union.
If you have any questions or you need some help, guidance let me know. I'm happy to help.