easiest way to open a UK current account when on fiance visa?

what is the easiest way to go about this?

I do not have a proof of address as my fiances mother is sponsoring us, so how would I be able to show proof of address?

would a council tax bill of her mothers along with a letter written and signed by her mother saying I am living there be sufficient along with my passport?

No, you need proof of address in your own name.

what is the easiest way to get this? is there anything that would take just a few days or is my only real option to try and get something from my US bank account sent ?

You probably need your spousal visa, but you could apply for a provisional driver's license. I don't think there's any proof that will come in a couple days.