Okay, I believe I have cracked Tommy Quickly, and it looks like he's still alive.
So stumbled across this post, where there's a comment from November 2017 saying that Quickly was alive and well in Skelmersdale, Lancashire.
Pump this into Ancestry and you get:
MR. THOMAS QUIGLEY (note – no middle names/initials)
BORN 1945–1947 (note – matches the DOB commonly used for him)
RESIDENCE 2003–04 SKELMERSDALE, LANCASHIRE
I'm certain this is him. There are only about a dozen people in the UK born in that decade with the same name. Since 2003, none of the ones who have died fit the profile of our man (i.e. the birth years are too different, or they died in a location which doesn't fit what we know of him).
There's also this, it's an amateur fan page on Facebook but it does talk about him in the present tense and they did put up a post in July wishing him a happy 75th birthday.
One more thing I will add – a common story online is that he developed a drug habit and later fell from a ladder which caused him severe brain damage. This isn't new – it's printed in the books I've been using to compile the UK Singles Chart info (they're from earlier to mid 00s). Now, these books are full of rubbish in places but let's suppose it's true – if it had happened at a young age, there's a good chance that he survived and lived out his life in some degree of isolation, maybe with limited faculties. It's entirely possible, and would account for why he has been "missing" for decades.
Personally I think the drug problems and fall might be true. I think the "severe" brain damage part may be an exaggeration. If somebody has commented saying he's "alive and well" then I tend to think he is?