Unless you're a Cockney Villain gangsta type in a Hollywood Movie or from "norff or east Landon" or its suburbs, no one in the rest of Britain really pronounces it Bri'ish. We don't all wear bowler hats, live in London, live in quaint little thatched cottages, know the Queen or speak like Mary Poppins.
The Queen most certainly likes her "T" to remain a BriTish institution.