A cross city route that is mentioned was most likely the 29 or 29A. The other routes south were not cross city routes (as far as I remember).
Astonian and the Old Mohawk, I think, have the northern route more or less correct. I only used the route in the north of the City to get to the Perry Barr speedway track.
The southern section (from Snow Hill southwards) went along Steelhouse Lane, around the islands (General Hospital and Central Fire Station) into Corporation Street, Bull Street, High Street, Bull Ring and then on through Digbeth, Deritend, and onto the Stratford Road. The 29 continued along Stratford Road turning into Highfield Road but the 29A had a more circuitous route along Wake Green Road, Sarehole Road, Robin Hood Lane, Keddleston Road (where two buses did not pass easily owing the the narrow road), Scribers Lane and then finally Baldwins Lane. This is how I remember it from 1945 until 1954. Bearing in mind that this time period is over sixty years ago there may be a discrepancy.
From the southern terminus the route in the northern direction followed the same roads until after climbing the hill at the Bull Ring. The route then turned left into New Street, then Corporation Street and Bull Street (Edward Greys store). It then continued down Snow Hill along the northern section described by others.
During WW2 many bus routes were diverted along different roads to their normal route. This was due to bombing (craters/rubble) and other issues. The 29A was rarely affected but inner city routes were; particularly the tramcar routes. I gather many routes were so disrupted due the the enormous amount of re-construction/re-development that took place after the mid 1950's.
Mention is made of the new book by Malcolm Keeley. I have a copy, there are a few here that also have and one stalwart in The States even has one. lol