With the American job season warming up and several jobs posted already for the fall, notice the several British philosophy jobs one can apply to. The UK academic job market does not operate on a timetable; there are jobs all year round. I've been on both sides of the interview table in the UK, and here is my take-down of the British interview process. For another, more comprehensive survey of the British philosophy job market, see here, and for a more general review of the European job market, see here. I refer to the Daily Nous post linked to earlier for terminology about the terminology for positions (e.g., lecturer = assistant professor in permanent position).
Before the ad is posted, the search committee decide on a set of essential and desirable criteria which will be specified on the job ad. The ad almost always appears on jobs.ac.uk, and also often (but not always, so be on the lookout on jobs.ac.uk) on PhilJobs and the mailinglist Philos-L, maintained by the University of Liverpool. Usually, a job needs to be advertised for a month or longer before the deadline.
You may guess that essential criteria are indeed essential, for instance, if they need someone to be able to teach intro logic, it'll be showing up as essential in the ad, and there's no point applying if you don't meet the essential criteria. By contrast, desirable criteria allow more leeway, and if you don't meet all of them, you might stand a good chance in if you meet some of them.
Applying for a British job is quite a minimalistic affair. You rarely need a research or teaching statement; you only need a cover letter, a cv, and a writing sample (in some cases, a writing sample is not initially requested). In some cases, you need to reformat your CV in the peculiar format of the HR website of the university you're applying to, which is a hassle. You typically need 3 reference writers, who don't need to send a letter unless explicitly asked in the ad. You provide their contact details, and they will be asked for a reference if you make it to the longlist. In your cover letter, specify how you meet the essential and desirable criteria.
The process is quick, compared to a US job. The search committee have a very short time to skim through all the applications. When I was on a search committee in Oxford (for a postdoc position), we had about a week to go through over 100 applications. I've heard for a two-year visiting position that the search committee had only 5 days or so to decide on a shortlist, all of this on top of teaching and the significant administrative duties UK academic shoulder, to decide on a shortlist. From this it should be evident that clarity and (relative) brevity are key: do not send more materials than requested, and keep things clear and succinct. Do not expect that people will dig in your CV to find hidden treasures.
However, if a writing sample is requested, send something long and substantial rather than a brief piece. This may contradict my earlier advice about being brief, but typically writing samples are only consulted at the long-listing stage (when about 12 applications are withheld), so the SC will not read many of them. Longer writing samples are better because of the REF*.
If you have not heard anything within a month, you were probably not longlisted or shortlisted (sometimes there is a two-step process where writing samples are only requested & read once the SC has decided on a longlist of twelve or so).
The interview process takes place in one single step: about five or six shortlisted candidates are all put together in one day, to present their research, teaching qualities and other relevant skills in a jobtalk and an interview. I'll discuss these in turn. Being all together on the campus on the same day is awkward. Often all candidates have lunch together, sometimes in the presence of some of the search committee members. Do not be put off by the competition! At this point, all candidates are very strong. Just try to keep your head cool and remember that as an outsider, it's very difficult to gauge "fit" (people sometimes lose their cool because a candidate seems perfect for the job, and they then subsequently hear that person did not get the job).
The job talk should be usually pitched at intelligent undergrads. These are sometimes present, but you can usually expect the audience to consist of the search panel, and a few interested graduate students and faculty members. Given that search committees are quite broad, often consisting of people who are not philosophers (next to philosophers of course), but e.g., people in classics and economics, it is vital your jobtalk should be a topic that is understandable for non-specialists. It can, and usually should, be something on your research and so it can be specialized, but keep in mind that people who aren't philosophers should be able to see why your research is relevant and important. To display scope, make sure your job talk and writing sample are different from each other. For one thing, people may ask you questions about the writing sample in the interview.
The interview is conducted by a panel of about 4-6 people, usually 2-3 people from the philosophy department, and some others, including the head of the school (which may include subjects like religious studies, history or languages). So, suppose your job is in ethics, you might have the other ethicist in the department, a philosopher of physics, a scholar who studies Chinese 19th century history, someone in chemistry, and someone in music. The philosophers might be the only ones who can gauge some of your answers about research, but the other people have a large say in the process (the one panel I was on was for a psychology postdoc. I was a philosopher, and the job was a very specialized neuroscience position. In spite of my lack of specialist knowledge, I did have a full say in the discussion when we decided whom to hire). So you need to keep this in mind when answering questions!
Usually, the opening question is "Why do you want to work for us?" - that question provides you with the opportunity to show your fit for the job, and that you've done your homework. You need to know about British and European grants you would like to (and should!) apply for. Next to the department, you need to do your homework about the university more generally (people often ask whether you'd be able to collaborate with people outside of the philosophy department - you should have an answer ready for that). You need to be prepared to ask questions about "pastoral care" for students (i.e., how to help students who are struggling academically or in other respects). You need to read about the REF and show confidence you'll be able to submit 4 high-quality outputs by the end of 2019. Weirdly, people almost always end the interview by asking "If offered the job, would you take it?" It seems prudent to say "Yes" (unless you have genuine doubts.)
The offer is usually extended on the same day or next day. If you haven't heard anything within 24 hours, prepare for a depressing agonizing wait for the impersonal HR letter that says you didn't get it. I say "agonizing wait" because occasionally it does happen you hear something later, for instance, because choice #1 didn't accept the offer (it's not uncommon to use offers as leverage at one's present institution) or because they need to jump through some HR hoops first before they can tell you. Still, not having heard after 24 hours means a 90% chance that you were unsuccessful.
Once the process is completed, it is useful to ask for feedback - SC members make notes, and they can offer advice about how you can improve your chances in a next UK interview (also phrase your request for advice in this way, rather than "Why didn't I get the job I so clearly deserve").
*I'm not going into the REF here, but briefly, it's a method of assessing and ranking research at British universities, taking place every five years. Each faculty member who is employed above a certain threshold (if I'm not mistaken, who has a .2 appointment at least) submits 4 high quality items, usually monographs and articles, which are assessed by a panel and given a number of stars, from 4 to 1 in terms of how world-leading, original, significant etc they are. The next one is in 2020. What I've observed is: in the time leading up to the REF, British universities aim to lure people who have published well from other universities with attractive offers; some universities give American world-leading scholars .25 appointments so as to game the system (their 4 outputs would typically get high scores - it's a win-win for the American scholars who get a nice summer income and for the university who gets a good REF score, but it's misleading for grad students who might be attracted by the names of famous philosophers who in reality are hardly ever there - just my cynical take).