you should go to the careers center, and also start talking to vcs - there are several in oxbridge. also, your college will have contacts with alumni and will happily put you in touch. alumni are an easy way to get hired.
I think your real problem is a lack of direction - but fear not. this is the point of going to a good university, in that you have access to almuni contacts who will happily guide you.
there are frequent meetups and events for oxbridge, and in london there is the club that is well worth the money to join to access the networking (although perhaps not so useful at this particular moment).
join the club, reach out to your college admin staff, and in both cases find people in a wide range of industries, present yourself as a recent PhD grad interested in applying the rigorous research and critical thinking skills you've acquired in a more practical manner.
you can really apply for anything - so you need to work out what you're passionate about. the book "what colour is your parachute" is very very good for this (skip the first half, it is the second half where it shines). you should also read around on venture capital, and apply the same ideas founders use to to vet/sell ideas to determining what you want your career path to be.
for example, "traction" by the guy who built duckduckgo. just keep in mind you're getting your brain ready to think unconventionally for jobs. don't waste your time applying via the online systems, again, you paid money to go to oxbridge, so use the connections to get a job.
oxbridge don't really train people to think commercially - ie selling yourself and putting the money value of time above all else - i think because they have such a strong reputation they've coasted on that (fairly or not!), while american universities have to be much more focussed on selling skills, so you tend to come out of them a little harder.
anyway, point is, you're oxbridge, welcome and congrats! use the network, use the network, use it some more.