I agree with OPebble. The kind of service you're asking for is a specialist shoe full repair from the original maker and is intrinsically a service that requires time and care. Even if there were a Church shop at T5, they wouldn't be able to do a proper service in a short time. If you want a quick service, you may as well look for a standard all-makes cobbler, but I'm not sure even that is something that's very practical while you're waiting for a plane - doesn't glue take time to dry, etc.?

A New and Lingwood repair costs about £100. I imagine Church is about the same (although it's been a few years since I wore their shoes). It's not a service I'd particularly want rushed.