I am quite sure that a number of taxi companies exist in the area around Knutsford who would take you to a hospital in the circumstances that you describe above. They would call directly at your home address...no need to hobble to the nearest bus stop.
Given some of the horrendous bike crashes I've seen, you may be looking to dial 999 if you've been struck by a vehicle, although you may need someone else to do that for you...
One of the issues with cycling to airports, though, is that the approach roads are not always cycle friendly. Take Glasgow Airport - the main approach is from the M8 motorway. Similarly from Edinburgh, you've got the A8 dual carriageway (with a motorway at one end and a bypass at the other). Even Aberdeen sees the cyclist negotiating the busy A96 roundabouts and driving through a busy industrial estate. I'm pleased to see East Midlands have put some thought into it, but it's still not the norm.
I cannot think of a major airport which has good cyclone infrastructure that would allow people to realistically travel by bike. And there is realistically little demand. Most people traveling to airports (passengers) are going away for at least one night, so it's impractical to carry a suitcase on a bike. Why would Manchester Airport want to spend money on bike racks and cycle paths when it doesn't benefit their customers? I think it's more realistic to improve the public transport links, which would benefit employees and customers (even for 4am starts - airports can be surprisingly busy at 4am!). I recently took the 4am 727 bus to Aberdeen airport, and it's amazingly well used; mostly with workers at that time of day.