This kind of depends on what sort of PhD you wish to study.
For the most part, you don't get accepted for a PhD position in the UK by applying to a graduate program. Most universities/departments simply do have a formal "program". There are acceptations to this of course. Several of the larger funders run formal programs, look out for the Wellcome Trust PhD 4-year programs for example. Some of the research council's Doctoral Training Partnerships (DTPs) operate on this basis as well. These are rare, and tend to be highly prestigious and often associated with research institutes rather than university departments (the Wellcome trust ones are also very well paid compared to other ways of doing a PhD). They are recruiting earlier and earlier in order to try to grab the best students. For example, the Sanger Institute 4-year program's 2017 deadline with 2nd of December.
For the majority of PhDs in the UK though, the supervisor will apply for funding from a funding body for a particular project. If they are successful they will advertise individually for applicants, interview on their own schedule, and the successful candidate will start at the earliest convenience of the both the supervisor and the candidate. This often ends up being the start of the next academic year as the candidate generally has to finish their Undergrad/Masters degree first. When these positions are advertised/when the closing dates are will depend most of all on the funders grant deadlines. Our DTP (where we get most of our students from) has a dealine just before christmas and we will generally be advertising in Feburary. A good place to look for adverts for studentships like this is FindAPhD.com. Another place to look is jobs.ac.uk. I would have alerts set up on these sites for the whole year, but expect the busiest time to be the first quarter of the year.
Finally, if you will fund your PhD yourself somehow (either though your own funds, or that of a company or from a foriegn government), its generally up to you. The application process may take a couple of months and if you are not native the visa process can take months as well. If you want to aim to start at the beginning of the new academic year (which is not a terrible idea, since you'll be starting with all the RCUK funded students, making things socially easier), I'd probably start this process in the early spring: give you plenty of time, and it you are sorted early, waiting to start is not going to be a problem.