I had the same symptoms when I moved into a new house with a neglected combi boiler last year. The system was full of oily oxide flakes and magnetite sludge due to neglect (and presumably no decent inhibitor in the water for a long time). The heat exchanger was partially blocked with these black flakes, and the restricted flow caused the boiler to overheat the water going through for a short while, and then cut out until it had cooled down again.
The 'standard' fix would be to spend £300 - £400 on a powerflush to clear out the radiators and boiler, then fit a magnetic filter. My radiators didn't seem badly sludged up though, so I just fitted a magnaclean filter and ran a couple of doses of Sentinel X800 cleaner around the system. To get the flakes out of the primary heat exchanger, I added water to the system until the pressure was a bit bellow the the pressure required to open the safety pressure relief valve (on my system the pressure relief valve opens at 3 bar, so I was filling it to slightly over 2.5 bar). The magnaclean filter is fitted on the central heating return pipe by the boiler, so with both valves on the filter closed, I removed the drain bolt in the bottom of the filter. With the pressure at 2.5 bar, water still being added to the system (and the boiler turned off of course!), I opened the top valve on the filter so water went backwards through the heat exchanger/boiler at pressure and into a bucket under the filter drain hole. I repeated this for 20 - 30 buckets worth until no more black flakes were coming out of the heat exchanger. (I chucked each bucket of water into the bath to see easily how many black flakes were still coming out.) Then after putting everything back together and topping up with plenty of inhibitor I sparked it up and it was running fine and toasty again.
You don't mention your hot water. Is that working OK? If your hot water is OK, your pump must be working OK, and there's a good chance that your boiler can push enough water through the partially blocked primary heat exchanger fast enough to keep going when it is only feeding the Hot water secondary heat exchanger, but it can't push enough through the primary heat exchanger to avoid overheating when it is dealing with the whole central heating system?
You should be able to tell if the pump is working by feeling the pump itself, or feeling how fast the first slug of hot water moves down the central heating flow pipe.
If your hot water is also a bit tepid or inconsistent, your secondary heat exchanger is probably also partially blocked. They are fairly easy to remove (in my very limited experience) once you have isolated and drained the mains water and central heating water in the boiler (and again turned the boiler off of course!). Once it's off you should be able to easily see for yourself if it is full of crap, and if so keep flushing it under the tap until no more flakes come out again (start by flushing against it's normal flow first until no more flakes, then from the other side until clear, then against the flow one more time). If it's really bad, you can goggle and glove up and use a bit of acid based limescale remover on it.
Another possible cause of 'cycling' like this is a faulty three way diverter valve (that sends the heated water to either central heating or hot water system). You said you can feel the slug of hot water going down the central heating flow pipe though, so it's probably not that?
Good luck (it's too effing cold atm to be without heating for too long)
EDIT: I should probably state again that I am no expert here, and I have a different brand of combi boiler to you, so... I paid the call out for a professional plumber to come look at my boiler and confirm my suspicions, and I ran the idea passed him first before doing any of the above
2nd Edit: My boiler is an old Baxi Poterton Main Combi 24 he.
