Hi,
There's a number of topics within this section about this, so do have a read through of those as the basics are pretty much the same. Get a Pay as you go mobile phone contract, tesco club card or anything which is easy to get started with no credit history. purchase regularly etc and build up a relationship with those companies. You might also look into stumping up front the cost of a mobile phone handset so that the companies give you a monthly rolling contract and then later a full 12 month one. It's the longer term money managability the credit agencies score you on. Get an account with credit expert
http://www.creditexpert.co.uk/ and look at their various guides on how to build credit. Getting on the electoral register is a very good enhancement, but I know it's more involved if you are American (or foreign) and not a British Citizen etc.
If you have an American Express account in the USA, they'll gladly transfer your credit history internally to the UK offices and they will also begin to report to the UK credit agencies to help boost your rating.
There's more nuances recently as the financial markets and companies try to come out of the financial turmoil, one aspect I'd suggest is to get a pre-paid credit card, perhaps from PayPal, and operate that card for some months and then give them a call to see if they'll allow a small limit £100 or £250 say on a normal credit card from them. Another thing would be to perhaps buy something you can afford right now, but put it on a say 6 monthly payback scheme, interest free hopefully. Another aspect is that you are looking for accounts on your credit history which show as 'Settled' - this means you've had some sort of credit and have successfully settled the account.
When you see your credit report, you'll see the monthly markers system showing how many months you paid without being late etc. Things to watch out for are not to make too many formal credit applications and not to miss payment dates.
By monitoring you report each month and making changes accordingly, you'll start to build that history. It's tougher than it used to be since lending practices have been tightened, but be patient and it will work for you.
Cheers, DtM!