I heard you can open up one account, and open more accounts under those really. But it really depends on what bank. And if they let you do it. I know that some banks have a program or something out now days that they know if you open up too many accounts at the same time it's not good! Someone on the forums explained to a bank manager that they needed more accounts for private reasons and so on. They understood them and let them do it.
From forum postings, you have likely noticed that some banks in the UK are favoured more than others.
You could then decide to pick RBS, Natwest to start with. Now you check their sites, but both of those banks allow you to make online accounts AFTER the 1st account is approved. You pick making basic accounts rather than current. That way the bank is not running credit checks & you are not flagging your accounts at any one bank.
You would not rush into doing this at all.