First off the majority, I seem to recall all bar one (though I can't find a photo to substantiate that), were in a neutral livery with a dark blue tail with the legend "Super One-Eleven" across it and small BEA titles of the forward fuselage. Again, and excuse me if this was an urban myth, this was because they were used on the German internal routes between various western cities and Berlin for political reasons involved with flying into the divided city. Some of the fleet was rotated through Germany while being based, as you say in Manchester. I remember in my spotting days seeing -510s at LHR, not sure where they arrived from, but I assume they were MAN-LHR services among possible others.
They certainly operated the BEA scheduled network from Manchester but also appeared on some BHX routes, notably DUB-BHX. Again, I may be wrong, but I can't recall any of the -510 fleet being based at BHX. The first BA 1-11s based at BHX were the former Autair/Courtline examples G-AVGP and G-AWBL and former American Airlines -400s, G-BBME/F and G. These were joined by three of the last -500s to be built in G-BGKE/F/ and G.
I'm sure @rog747 can flesh out a great deal more detail.