Car Owners Hub UK - Built by Drivers, for Drivers

4 door, 2.0 litre, 214k miles 2003 Chevrolet Tracker/Suzuki Vitara.

In the past, I've had problems with the engine internally burning a lot of oil. Moving when not in use. Idling at red lights caused the tachometer to occasionally drop to 500.
After two weeks of sitting, the car's engine suddenly stopped turning over. A thumping sound that repeats indicates that it is churning and struggling. The tachometer goes up to a peak and then comes down to zero. Its tone drops to a whine, as if it were exhausted and ready to give up. I haven't even waited half a minute after inserting the key.
Additionally, the lights work properly when you turn the key once to start the engine. However, it remains in its upright position.

Checked electrical system and battery, replaced alternator less than a year ago. I just changed the oil.

There is no clicking, but the starter and fuel pump are making me nervous. Engine might be about to overheat. Have you any opinions?

Perhaps you could record the engine in action and post the link to it in this thread so that we can all hear it. You could use YouTube or another video hosting site. It could be a problem with the starter, wiring, ignition switch, timing, etc., if the battery voltage checks out.

We must be clear - for the majority of us, "turn over" means the engine (the crankshaft) literally turns over; when you turn the key, the starter engages and the engine turns over; it might or might not fire, start, and run.

To me, it sounds like it's turning over but not firing because you can hear it churning and struggling, even though you're saying it won't turn over. Based on your description of the sound changing to a whine, as if it were exhausted, I would assume that your battery is weak or dead.

A minimum requirement for remote diagnosis is an on-site informant who is knowledgeable enough to describe the situation or, as Bex suggested, film it, since we cannot see or hear what is happening.

In my opinion, the engine is completely broken. I agree with what you're saying:

Try a compression test if you are able to. In that case, empty the oil and search for "shiny bits"; if that doesn't work, check the oil filter; it appears like you've broken a piston and (part of) a connecting rod.

Make a video and share it. We can only speculate at this point.