This is exactly why I stopped worrying about delays after switching providers. My current independent taxi company for Airport Transfers (3Cat) tracks the actual flight number, not just the time I booked, so when my flight was delayed by ninety minutes last year the pickup just adjusted automatically. Didn't have to call anyone or explain anything. Worth checking whether whatever service you're using actually does real flight tracking or just works off a fixed time, that's the detail that matters most here.