I booked a home COVID test online on 24th September, it arrived on 25th, delivered by Amazon. I returned it the same day and the result (negative) arrived from a lab in Glasgow on Monday.
The test was booked as part of the Kings College Zoe program, not because I had any reason to believe I was unwell.
IF the system worked everywhere as well as it did for me then there shouldn't be a problem, but clearly the availability of tests is variable. It does seem that a home test is easier to get, but it isn't an enjoyable experience: taking the throat swab makes you gag and want to throw up and the nasal swab makes your eyes water and is very uncomfortable. Caroline seems to enjoy carrying out the sampling... !