Who compensates my UK train ticket when the train in the ticket didn't exist?

After a little bit of digging, it looks like the problem you ran in to was indeed the responsibility of Southern Railways. Hidden on the relevant National Rail engineering works page it states

Services between East Croydon / Clapham Junction and Watford Junction / Milton Keynes will not run between Shepherds Bush and Watford Junction / Milton Keynes Central. Please note, these services have been suspended due to a shortage of train crew. Customers should use alternative routes. More details can be found here.

Customers should use alternative Southern service to / from London Victoria, London Underground services by any reasonable route between London Victoria and Stanmore, and on London Northwestern Railway bus services between Stanmore and Milton Keynes Central via Hemel Hempstead.

As MadHatter states in the comments, you should thus be able to claim compensation from Southern via Delay Repay.

OK, there's a few things here.

1) You shouldn't have had to have bought a new ticket just because they scheduled engineering works after you'd already booked. Your ticket should have been honoured on other trains (National Rail Conditions of Travel paragraph 28.2) on the day given you had what was a valid itinerary at the time which is now made invalid through no fault of your own. It's one thing if you actually saved money by rebooking (ie the rebooked ticket was actually cheaper than what you originally paid and you got a full refund from your original ticket), but it's quite another if you ended up paying more — this isn't clear from your original post. Though it gets a bit more complicated if you particular care about having reserved seats, I must admit.

2) Delay repay on trains removed from the system* is a little nefarious. Realistically since you did end up rebooking, I would for simplicity's sake look at claiming against what happened with your final booked itinerary. In this case, as others have mentioned, Southern removed the train from the system at short notice due to a shortage of train crew. It is NOT the fault of The Trainline, assuming (as seems likely here) Southern didn't remove their train from the system until after you booked it. Part F of the Conditions of Travel linked above clearly set out your rights when it comes to delays and cancellations, but it's never explicitly mentioned whether "cancellation" just refers to on-the-day cancellations or also trains cancelled further in advance. I think the most reasonable interpretation would be the latter, so I would assume that and work from that basis, personally. I would therefore submit a delay repay claim to Southern — if the web form won't let you (eg because you can't select the train you were booked on due to it having been removed from the system), send the claim by email instead. If they cause a fuss, you can escalate it to Transport Focus. I doubt you will be completely denied the claim, especially if you show them your original booked itinerary, but if you do, I would be reasonably confident that appealing to Transport Focus will ultimately prove successful.

* ("Removal from the system" is not an official term but I've seen it used here and there. It's really just another type of cancellation, but it's more like a schedule is cancelled rather than a train — the schedule is marked as CAN in the system for that particular day (or VAR with some stations removed, like happened here — this is your train, it was just curtailed to start at Shepherd's Bush). Being removed from the system happens routinely for schedules that (for instance) don't apply on a certain day because of engineering works. This is why the trains don't show up on screens at the station etc., because this process is intended for trains that never should have existed in the first place. The big difference is that this is supposed to happen before journey planners will let you book trains on that date, thus meaning that nobody should see that train as ever having existed — but for a variety of reasons, recently this has been happening after the deadline so trains are "disappearing" long after people have already booked on them — this has in fact happened to you twice on one trip, first with the Virgin train then again with the Southern one!)

You booked with Trainline so you should contact them. They have a complaints procedure and their T&C s5 https://www.thetrainline.com/terms refer to booking errors and compensation for delayed or cancelled journeys, Alternatively try claiming direct - National Rail publishes a summary of consumer rights and links to each of the rail operators http://www.nationalrail.co.uk/times_fares/ticket_types/121354.aspx

I have, in the past, claimed compensation for a delay from one company, who passed my claim on to the operator who was held to be responsible, who then compensated me. I guess this caused a small delay to my claim but, as I recall, it was all sorted out within a couple of weeks, which was plenty fast enough for me. This leads me to believe that you should claim compensation from any of the companies involved and they will pass your claim to the correct company.

_{The situation in my case was that I took a local train, followed by a regional train, followed by a long-distance train. Becuase I knew there were delays on the regional route, I took and earlier train than originally planned, so I could have my reserved seat on the long-distance train. As it turned out, all three trains were late, with the long-distance train being more than half an hour late. I claimed compensation from the long-distance company, since I had successfully caught their train, which I'd always planned to catch, but which was late. They, (unfairly in my opinion, but not my problem), passed the claim on to the local train operator, who compensated me.}