You’re in the UK? Airports like Heathrow have testing centres you can turn up early to and get tested.
Discussion
Apparently it's full vaccination or nothing, it explicitly states that a test isn't sufficient :/ My only hope is to get a doctor to sign an exemption but I've got 24hrs and even getting an appointment will be hard.
Try an online service like “Push Doctor”. You pay, but can do any hour, and because you pay they’re more likely to give you what you need.
Other services are available ;-) Just looked at their reviews and issue where they make you pay but not clear if appointments available. I’m sure there are other similar apps.