The larger intercontinental aircraft have a surprising range of medical equipment on board.

If something serious seems to be happening the CSD will call for any medical experience from passengers. Almost unknown for no one with doctor/nurse experience to be on board, by the way.

Flight deck will contact their operations centre and be patched through to Medilink, a worldwide organsiation with a 24x7 response centre based, I believe, in Phoenix, Arizona, and staffed with aviation-knowledgeable doctors. In conjunction with on-board doctor and flight captain they make the decision about what to do.

There are some interesting rules. From Johannesburg to London, over land up the length of Africa, major airlines invariably refuse to put down a medical emergency anywhere between Johannesburg and Tunis, on the Mediterranean coast. Facilities just considered to be not good enough. They may do a substantial turnback to get to these places if needed.

En-route diversion can be very costly, getting up to £100k if crew go out of hours, passengers have to be put into hotel, subsequent flights for the aircraft are disrupted, etc, so is only done if really necessary. Having said that intercontinental airlines know what they are occasionally in for, that they are carrying humans and the percentages of times they need to do an en-route diversion. It's just one of the costs of doing business as an airline and is all allowed for in the budgets.