Google shows a rather circuitous route to the far entrance of the station. If you follow the route that I described via Temple Street, it should take 5-6 minutes. And, on reflection, I’d definitely get off at Snow Hill and walk down to New Street rather than stay on until Moor Street.
The train is quicker than the tram. Moor Street station is virtually next door to New Street station. The train stop before Moor Street is Snow Hill. It will be almost as quick to get off the train at Snow Hill and leg it across the City Centre via Colmore Row, St Phillips Square and Temple Street.