Working for the same company as Shadowpurser etc, my experience of Medlink is very positive - if only for reassurance etc.
On the shorthaul fleet though, a lot of aircraft have no HF comms so all you are left with is ACARS, with about 120 characters of message space which makes it very difficult to compose meaningfully. Then it takes up to 45 mins for a reply. I literally spent about half a flight from LHR to IST on ACARS to Medlink - waste of time, much better to throw it at the deck it the problem deteriorates.
On longhaul aircraft most have Satcom facilities (or at least twin HF with Speedbird London on call to patch through). The satcom facility is excellent (albeit v expensive) and you're straight through to Medlink. I have used it several times and never have I seen it refused when the cc decide it is necessary - it costs nothing (fixed annual subscription). Have also got the cc member to do the talking to complete the loop - also very useful experience for them. Recognise however on a flight from NRT to LHR (for example) that even a life threatening scenario on board will probably not result in a diversion - EVER. In certain parts of the world there aren't satisfactory ground facilities to help the patient, and you're better of in the air travelling at 600mph towards somewhere better equipped.
So my take is that on LH Medlink should never be denied when requested, just beware of our often limited options. On SH it is of much less use.
BTW, don't expect any on-board doctor to be allowed on the FD to speak directly to Medlink!