The first thing you need to do is to open a bank account. Although a letter from your former bank will be useful, I think it will be far more important to have a letter from your employer stating your salary and as much detail about your job as possible (salary, if your relocation is short term or long term, etc.).
When we first arrived here Citibank was very sympathetic and gave us a bank account straight away. HSBC is constantly advertising itself as your local global bank, so it may help to go to your local branch to see if they live up to their own hype.
The second thing you can do is register with your local council. If your landlord has not advised the council, then simply give them a call, explain your situation, and they should arrange everything for you.
Something else that applies in England (and I suppose in Scotland, but alas, they are different countries really, so beware) is the need to register in the electoral roll, which is checked as part of granting you any credit, try this website for this: Directgov: Apply to register on the electoral roll
As you may know, EU citizens can vote in local elections, so you can actually register in spite of being foreigner (we, non EU citizens have this eternal problem of not existing in the electoral roll...).
As for mobile phone, you can get a SIM card (they dish them like confetti) and an unlocked phone, then you can buy airtime as needed, but I suppose once you are settled, registered in the electoral roll and with a new bank account, you should be able to get a regular contract.