For going to work in the computing sector, "brand name" of your university means essentially nothing compared to having good work experience and/or a good portfolio of self-directed projects which can form the basis of a portfolio on github or similar. Even aside from that, for CS specifically Southampton is quite prestigious in academic circles (for whatever it's worth). I'm not really aware of Durham's CS programme being particularly notable, other than it being offered by Durham which has some other notable departments.
Ultimately your employability will depend on you, not which university you go to, however I'm very skeptical of your belief that Durham CS is more employable than Southampton CS even setting that aside. That sounds more like league table obsession/Oxbridge wistful thinking of wanting to go to a collegiate univerisity, than anything grounded in reality. I would suggest you examine your reasoning for coming to that conclusion carefully, and evaluate what evidence, if any, you have to support it.
At the end of the day, you should go to the university you think you can be most productive at and think you will get the best degree classification from, and ensure you proactively seek out work experience opportunities and work on (and document) projects outside of your studies wherever possible. CS generally has quite poor employment prospects, to the point that the government ordered an inquiry (I think actually two, but I only have read the Shadbolt inquiry) into the poor prospects of these grads.