I will do this on the day if anything happens.
The reason I asked this question is because I expected the answer to me "yes, take the first train"; nobody would notice for a short 2 minute journey anyway... however, is the answer technically that you have to wait for this train unless you will be over 60 minutes late/cancelled by which point you gain the ability to be rerouted?
Theoretically (and will not attempt) would you be entitled/expected to delay yourself waiting for your booked Thameslink, gaining delay compensation in the process? This is nonsensical for a 22tph core section.
Before GTR made Thameslink trains "reserveable" for the purpose of "counted places", it was absolutely unambiguous that you would take any available train.
However, GTR have muddied the waters; this could be avoided if only Rail Delivery Group would formally say that the "connections" can be used on any "appropriate" service regardless of reservations, but I doubt they will do that.
You could ask GTR to see what they say, but I doubt you'd get a good answer.
However it's a completely moot point because almost no-one ever checks tickets on these trains, and even if they did, it would be the absolute pinaccle of "jobsworthiness" for anyone to claim that you couldn't take the first available train.
Next time, if you want to avoid any potential for any ambiguity, you could specify avoiding Thameslink when you book, which would give you an itinerary on London Underground from Farringdon to King's Cross St Pancras, but if it was me, I really wouldn't worry about getting on the first train that pulls in.
Tip: the escalators from Farringdon (Crossrail) platforms are near the front of the train, in the westbound direction, and the escalators take you to the northbound Thameslink platforms; it's a much easier interchange to make than the other direction. You then want to be relatively near the front at St Pancras (maybe about 3 coaches back?) for a speedy exit.
If you ever want more interchange time, e.g. if your arrival time is critical, you can specify additional interchange time at your desired location.