(Computer Science) Birmingham vs Newcastle vs Manchester vs York vs Southampton??

They are pretty much on a par really and very much depends on where you would like to be based. The Universities all have good employability but consider if they offer any year in industry as these experiences really help.
Marginally, Southampton and Manchester pip the other three, but really not enough to overide module combinations offered or city/town experiences. So really weigh these things up too. Also consider student experience (and it doesn't follow that it is the best at Unis with the 'best' reputations for looking after undergraduates) which you can find here:

https://discoveruni.gov.uk/

They are all Russel groups so therefore they all have a good reputation and offer good graduate employability.
If I was to rank them I would say:
1. Manchester/York
2. Southampton
3. Birmingham/Newcastle

It really doesn’t matter though, they are all good. I would also consider which city you would prefer to live in, do some research on that as that’s quite an important factor.