It's less than 10 minutes walking from Charing Cross station, just go through Trafalgar Square.
I use http://www.traveline.info/ to find out travel information.
It says that if you walk back to Trafalgar square you can catch the number 11 bus to Victoria coach station (23 minutes).
Not sure how long it takes to lodge your application, I think it a first come first seved basis, if not busy then it may not take long?
