Andy,
I think that Scroggs' final comment is worth considering as this is the route I took and it doesn't seem to have caused me any problems so far.
I studied at Uni for an Eng degree after which I got employed by BAE SYSTEMS as....yes you've guessed it.....an Engineer!!! I personally think that when I move onto a flying job, my background in aviation engineering should stand me in good stead. But thankfully for me I found job openings in Flight Operations whilst working as an engineer, and this then allowed me to change to a shift pattern (good for flying) and submerge myself with operational pilots and alike for help, guidance, and most of all, contacts for when i get into the sky fulltime!!
So in summary, I would do your degree first and then go at it full speed. Again, as Scroggs says, your degree would also serve as a back up should anything go wrong.
Hope this helps
ADRH