The timings and process depend on your College. Some Colleges have plenty of accommodation, some don't have enough. Check what your College says about this, or there is a central Uni web page that shows accommodation spaces and student numbers, or read the MCR website for your College.
If you want to live out, you can just chose to and make your own arrangements on the private market. Generally not considered the best option for 1 year MPhils or 1st year PhD but some people want to.
Otherwise, you wait for your College to make contact. The timing of this depends on a number of factors - what their situation is with UG v PG accommodation, ie easily interchangeable or quite separate, the amount they have, and whatever ballot rules previous students have asked for.
Mostly, nothing material happens until July or even late August, as most Colleges have to sort out UGs first, and results only come out in mid August. So you will likely not get told what accommodation you've got until after that. Some Colleges don't even ask what you want until then, some ask what you want much earlier, but then don't do anything with the info. You usually don't find exactly which room you've got until you turn up at the start of term.