Hi Jolyon,
It all depends where you are changing and whether you have to go through a gateline or two between trains.
At Farringdon, everthing is within the same gateline so there is no need to touch in/out, unless you end up going the wrong way which will mean touching out, crossing the road, and touching back in. The two journeys will be joined together unless you take longer than 10 minutes to cross the road (you won’t). If you follow the signs for Thameslink at Farringdon you shouldn’t end up at the gates. There is no need to touch either of the validators between the Elizabeth line escalators and the northbound Thameslink platform – they are only needed if you are using a paper ticket on Thameslink.
At Woolwich/Woolwich Arsenal you will have a short walk between stations, but again if you do it within 20 minutes the journeys will be joined together. I think I’d go the other way from City Airport to Canning Town and then to Custom House on the DLR. At Custom House you must touch out at any of the validators above the DLR station before touching in through the gate for the Elizabeth line.
In short, yes the system will join together the bits of the journey so you’ll only pay for one trip.