From my experience, the London firms are more scheduled in terms of accommodating for non-law graduates. From memory, I know Ashurst recruit on a rolling basis (friend of mine got a TC offer as early as November-ish), although he had, by that point, graduated a few months earlier.
Best thing to do is draw up a shortlist of firms you'd like to apply to now and see on their website if they have an earlier deadline for people like you, or if they recruit on a rolling basis. If unsure, contact the firms.
On a side note, be prepared to convince them that you are interested in a career in law.