Is it not the individual healthcare professional who makes your appointments? The nurse always rings me and asks when she can come – I prefer to be her first call in the morning so that the rest of the day's free, but she'd do evenings if I asked. Maybe try saying when they call that you're not available until 1700? Some of the participants must have work commitments, and I bet they're able to work around them.
The nurses in the Midlands have to deliver all their completed kits to a testing centre in either Sheffield or Birmingham by about 6pm, so if it's the same where you live, it may be the reason they're trying to book you in earlier. But there's no reason it couldn't be included with the following day's batch if the test's done in the evening.