I think these days, it's the vocational masters you choose afterwards that makes all the difference. There aren't many careers in ancient history! And there aren't any jobs in academia at the moment. But ancient history is fundamentally a fantastic subject - it's so interesting and varied.
Possible paths might be:
BA in AH -> PGCE -> teacher
BA in AH -> MA in Heritage Management or Curatorship -> career in museums, heritage, National Trust
BA in AH -> MA in Archaeology -> career in archaology
BA in AH -> MPhil in International Development (you could spin the application alongside whatever specialisation you take in the BA for the thesis, i.e. Ancient Near East, Egypt etc.) -> career in government
Or after BA you (or your child if you have child who wants to be an ancient historian?) could do a MA in whatever or go straight for any one of the many many careers in the civil service - https://www.faststream.gov.uk/diplomatic-service/index.html
A first class honours degree, particularly in a decent uni will take you far, no matter what it is in in, so long as the NEXT step is chosen carefully.