It might help to explain what exactly a flight plan is.
I'm not an ATC....just a lowly PPL, but have gotten fairly used to flight plans by now as I need them for pretty much every flight that I make. Like you the system in the UK came as a bit of a shock to me
When you file a flight plan, they get an AFTN message giving most of the details on your flight plan. (The stuff at the bottom such as life raft, life vests, survival equipment etc isn't transmitted to ATC).
ATC at your destination take this as nothing more than an intension to fly.
When you depart you get your flight plan opened/activated. All this really does as I understand it, is send a departure message to your destination airport with your take off time, and (I think) a reminder of some of your flight plan details included your estimated enroute time.
ATC at your destination now know that your intended flight is more than just an intension. They know that you've actually taken off, and they know roughly when to expect you. But that is it. They have no idea of your progress along the route, or your current location.
When you eventually arrive, they close your flight plan, which basically means that they send an arrival message to your departure airport stating your time of arrival, so that they know that you've arrived safely.
If something happens to you along the route and you divert, you are supposed to get a divert message sent (to your destination). They then know that you're not coming and bin the flight plan and departure message.
If something happens and you crash/ditch/disappear, ATC at your destination are under no obligation to do anything in the UK. However, assuming that they haven't split their coffee over everything, or ATC is closed and the airport is now uncontrolled, assuming that they have time to think ....etc etc...they should notice that your expected arrival time has come and you still haven't arrived.
I imagine that most don't get too worried at this stage, but if it goes on a bit longer, AND they notice (because they are under no obligation to notice or keep track of it) they will eventually start to suspect that you aren't arriving. They will probably start to check with ATC along your route to see if you're simply making slower progress, or to see if they can establish that you've diverted.
If all this comes to nothing they will most likely ask for S&R.
However this is all based on ATC being present, them getting your flight plan and departure message, noticing that you haven't turned up at the required time, having time to follow it up, and not having ended their day's work. They are not required to do this follow up, and therfore might not even notice that you have gone missing.
Even if they do everything like clockwork, it could be many hours after you've had a problem.
Imagine a 3h30m flight and you've problems and ditch 30 mins in. It will be another 3 hours before they are expecting you. They probably won't get too nervous for another 30 mins at least. Then they start to call around for other ATC/airports along the way...probably another hour's work, and only then do they ask S&R.
Now if your destination airport does not have ATC, then they are less likely to have your flight plan, to have received any departure message. Often in these cases, a nearby airport with ATC will get it instead. Of course these are less likely to notice anything wrong as they've no way of know that you've not turned up. They've no obligation (in the UK) to send the arrival message (because you've no obligation to close your flight plan). The airport that you've arrived at has no idea that you've actually taken off (because they don't get any departure message). It all becomes a lot looser.
Hopefully this helps you to understand the "it depends" a bit better, and to explain my comment that having ATC at your destination makes it a lot more likely that something will happen.
The other side of this is that your departure airport may well make enquiries if they get no arrival message, but probably not as they are used to things not being closed in the UK.
And even if it works, it can be many hours before anything happens.
Even outside the UK, it can take a long time. I remember one day having landed in Ireland from a internal flight, around midday, expected that my flight plan would be closed automatically because I landed at a full ATC airport (which by defination in Ireland meant that I had to have a flight plan). I forgot to turn on my phone after flying and didn't realise it until about 10pm. When I did there was a message from Dublin ATC checking to see if I'd arrived safely as my flight plan handn't been closed.
I was concerned that S&R would have been initiated as the message was from five hours earlier. When I called, to let them now that all was fine, I learnt that nothing had been activated. Perhaps if it was a cross water flight they might have been more concerned?
Anyway...I hope this is of help to you to understand what's going on.