China ATC restrictions not only cover flights between PVG/PEK, but all aircrafts passing through China Airspace....

CX flights from HKG to Europe occasionally get delayed by China ATC. There are quite a number of flights to Europe departing at around midnight. On some nights, you will see all these flights to EU got delayed by a few hours.

CX253 is an afternoon flight rather than in the mid-night where there is a wave of CX's 77Ws departure, there is usually spare frames in the afternoon and less likely resulting a delay like this due to technical.

1:50 is entirely reasonable to turn an aircraft. A marginally efficient ground staff, and CX is a lot more than marginally efficient, can get this done quite well.

Making a profit is generally acceptable in some corners of this world.

That appears to be an assumption, and probably incorrect.
CX253 was late, arriving 23:39 : CX254 scheduled for 22:20 departed 01:06 just 2hr46 mins late. Therefor, from your own account, 27mins were lost enroute accounting for an arrival 3hr13 mins late. You have proven that it was not the late arrival of CX253 or the late departure of CX254 which caused you to be late.
It was the "extraordinary circumstance" enroute which consumed the 27mins, perhaps adverse weather or ATC, in any event unless you can you can show the plane broke down it wasn't CX. If I were CX I'd consider my case watertight.
As an aside, you mentioned they transfered others to an earlier flight, I think that's called courtesy, an effort to reduce the number of passengers inconvenienced at no cost, a flyer on a codeshare award would rank lower with any airline than status holders and full fare paying CX customers. More than reasonable.
Times & data from flightradar

There's a bit of speculation as to if the HKG-LHR flight (CX 253) was due to a technical issue or ATC. I was on this flight. It was ATC. Immediately after the doors closed, the captain explained that the anticipated delay was approx 3 hours (which turned out to be 4 hours). The captain's estimate was based on the time slot given to the flight due to take off immediately before us, so that flight was delayed as well. He also explained it was necessary to get all passengers on board first before being able to request a push back time, and as such it wasn't possible for us to wait it out in the terminal. I was also a bit surprised that long haul flights were badly affected that day. The first meal service was done on the ground before take off.

Just a quick update on this. I've had a very positive reply from Cathay to my email. They have offered us 300 Euros each straight off in reply to my single original email. I had not actually even mentioned EU261 in this email or the length of delay. They have sent a pair of payment forms that we need to fill our bank details in on and email off.

So, top marks to Cathay. Seems like there was no argument to be had. Thanks to those posters with positive contributions to this thread. Hopefully others will have a similarly painless experience in the future.

Thank you for your feedback and great (and surprising!) to see that CX did manage this in such a positive way :)

I'm interested to know how you termed your email if you didn't mention EU261 compensation. I'm surprised they jumped straight to payment of compensation. But well done on getting the right outcome.