I'm currently trying to get my driving license in the UK, as a US immigrant. This means I need to go through the same process as a new driver. The wait was about a month between application and approval for my provisional license, then 1.5 months to take a theory test, then 2 months to take a practical test -- but that was choosing to drive four hours to a test centre with a test available, and if I'd wanted to take one in London, it would have been ~6 months, which is long enough that my American license would have expired and I'd be left unable to drive at all in the interim. (There's still a decent chance this happens, because if I fail the test, it's a month before I can reschedule, and then I probably have to schedule multiple months out again.)

Wait times were pretty bad in the US, too, before we left last year. It's not a government thing, but we had foundation work done on our house last year, and the lead time was about eight months between signing the contract and the work being carried out. Permitting alone took several months for what was relatively straightforward work on a single-family dwelling.