How fast is the admission procedure for single-slot PhD applications in top universities in the UK?
It can depend on the time of year. At my old UK university,the standard PhD admissions process would take over a month. If the timing worked out, we would generally lump single slot applications in with everyone else so that interviews and decisions happened at the same time. If the timing did not work, we would do them individually. This meant it was really down to how fast the PI wanted things to go. The PI would do all the leg work, screening, and interviewing, and then put a single candidate forward to the admissions committee. A week would generally be very quick, but a month would be on the long side.