Kings Norton train station is on the direct line to Sutton Coldfield - from there it is a few minutes walk up to Bishop Vesey's.
So it is a reasonable journey in my opinion, no changes and pretty regular train services. There are boys who travel from there - and also from Worcestershire - to get to Bishop Vesey's. The train will have a good number of other BV boys on it and the school told me at open day that the boys tend to walk en masse from the station to the school in the morning, and vice versa in the afternoon.
Entry to all the Birmingham Grammars is based on test score, not distance. There are no catchment areas.
The only time distance would be looked at, is when a number of pupils were in a tie for the last place available. In that case, it would be allocated first to any pupil in care, then on distance to school.
In general, though, distance is not relevant.
That said, your closest Grammar would be Five Ways, then Camp Hill. As you'll probably be aware, these two schools have historically required higher test marks to gain entry.
The Warwickshire Grammars do have a catchment circle. Priority is given to those living within it. If you're thinking of Alcester Grammar, you can check with Warks admissions whether you are in the catchment circle.