Switching just after moving home

-Selene

I've just moved from City A, where I lived in my own flat, to Village B, my mum's house. Currently I have three current accounts with NatWest -- one for income, one for bills, one for savings (I know the interest rate is terrible; I set up the account with NatWest intending to switch it to a high-paying current account but due to circumstances beyond my control wasn't able to do it earlier) and intend to switch two of them.

Account 1 -- currently receiving income but I hope to get the income paid into a new account to fulfil the min pay in requirements, and use this one for spending money only.
Account 2 -- expenditure. Looking at switching to Halifax Reward for the £100 cashback and £5 monthly. Was thinking I'd get income paid into this account (to fulfil £750 minimum) then transfer spending money out to NatWest.
Account 3 -- currently have savings of between £1-2k. Hoping to switch to First Direct for the boosted £150 cashback (may need to get £125 instead if I need to go in branch but still good) and then if/when my savings increase, switch to a better-paying account.

The problem? I have no UK passport or driving licence (DVLA needed me to send proof of ID last time I applied, but I was unwell at the time so they refunded me the money and cancelled the application). Therefore I need to be on the electoral roll which I am... in City A. I've applied to register at Village B but doubt this will be processed before the bonus deadlines. I also need to submit benefit letters and utility bills as proof of ID -- all with my address in City A. Gas and leccy are currently prepaid so no bills, water/CT in mum's name, she owns the house so no tenancy agreement etc. If I supply the bank with my utility bills etc with the City A address and explain the circumstances, would it be likely for them to accept those as proof of ID? I know I can go in to ask but the nearest First Direct branch will cost me nearly £7 in train fares so I can't just "pop in" if they're going to be 100% inflexible.

So basically I'm just wondering if anyone has tried to switch just after moving and how they managed it and if anyone has ideas for how I can get bank-appropriate proof of ID quickly it would be great (unfortunately Halifax at least don't accept mobile bills).