Lmobbshill wrote:... I am completely perplexed!
You have every right to be! You covered my immediate thought with "Both Churches are of the same denomination".
My guess would be as KSouthall suggests - family reasons for wanting the Birmingham baptisms - maybe it was their old church? And Notts because it was their current parish?
If this was the full service in both places, then in fact, so far as I understand it, it shouldn't have happened according to Church practices - you don't ask God to accept this child twice, He heard you the first time! However, there is a point in the Anglican Order of Baptism where the priest asks if this child has already been baptised. If the answer is yes, then the order of service for that child skips forward, looping round the actual baptism. This question and loop forward is there to accommodate things like children being privately baptised at home. Quite whether the name of the child should
be recorded in the baptism register in such a case, I'm not sure, but it does seem to happen. So it could be organised quite correctly with the priest that the 2nd service follows that pattern of the ceremony without the actual baptism.