What I idid was open a 2nd current account with my own bank, set up 2 direct debits £1 each to my daughters saving account, ( can even set it to your main account), then look at new account, (leave it a month or 2) so not looking like ur applying willy billy on your credit score
Check the banks offers
Some are os6 £1000 in to account within 3 months... ( could transfer £100 in and out 20 times)
Pay £500 in per month... transfer £100 in and out 5 times... as long been sat in account
Generally first direct, co-op, tsb, are worth look8ng at x