You can apply for pupillage at any time, though the majority of pupillages are advertised and offered during the period in which the pupillage gateway operates, which is from the beginning of January (though advertisements are put up and applicants and view the site before that) in any given year until offers are made at the beginning of May. As someone who has just completed the Bar course, I would assume that you're already aware of the pupillage gateway, and I'd hope that you applied for pupillage during this cycle. Either way, your next opportunity to do so in the main will be at the start of next year, so I expect you'll have to find employment in the interim to remain in the country on a work visa. There are sets that recruit outside of that timetable, but it's a small number compared to those who recruit during it.
I'm afraid I don't have any in depth knowledge of the immigration issues. It may be that there are others who can help with that.