There are quite a few non-legacy bank accounts that you can apply for a company account with in a very painless manner. Monzo, Starling and Tide spring to mind. I use Monzo as part of their business banking trial, but I previously have used Tide and (unfortunately) RBS.
Most company formation agents offer a free "bank introduction service". I have no idea how the logistics work, but I was able to open an account with Santander in a matter of minutes.
