If it were me, I'd wait until I get the formal letter - so you can make sure that all the Ts&Cs are what you're expecting.
There's nothing to stop you telling your consultant tomorrow that you've been offered another post which is more convenient for you, and that you are waiting to give notice (i.e. give him a bit of advanced warning) - then hand in your formal resignation once you have the offer in writing.
In the current climate, I wouldn't do it otherwise (tbh I'd always wait for the letter anyway).