Hi Dave,
Please look again at the leodis.org website, which has 3 photos of the Bull & Mouth, & a very brief history.
Technically, I agree that the pub was on Briggate, rather than Westgate, and I cannot understand why it would have been recorded as being in "Westgate", since the part of Leeds known as "Westgate" is a little bit further away. However, there was, from the checks I've done on Leodis & other sites, only the one pub of that name in Leeds, even though it had another name earlier and at some later point in the 19th century became the Victoria Hotel. It was finally bought by Woolworths in the 1930's and demolished to create an extension to their Briggate store, which itself closed as Woolworths in 1987.
So, if you are certain that the pub was in Leeds, this seems to be it.
Best of luck, Phil.