Sorry my initial post was vague on detail. Here's an example, although with Peterborough rather than Stevenage as the change point.
Your Outbound Journey, Friday, 13 December, 2019
Doncaster Peterborough
1 hr 3 mins
12:25 13:28
London North Eastern Railway train service to London Kings Cross (details)
wait for 26 mins (info)
Peterborough Finsbury Park
1 hr 13 mins
13:54 15:07
Thameslink train service to Horsham (details)
On arrival, go straight to the tube
Finsbury Park London Euston
16 mins
Take the tube from Finsbury Park to London Euston
Ticket
Doncaster to London Undergrnd Zone 1-2
Advance Single (BUS) £49.50
Only valid on booked London North Eastern Railway services and required connecting services.
This is the public (undiscounted) fare.
Restriction GC applies
Question
Most people would prefer to stay on the LNER train to Kings Cross, so does that ticket allow doing so?
It's not intended to be used on LNER from Peterborough onwards and the reason TrainSplit has put you onto the Thameslink service from there is likely because it enables a cheaper tier of Advance to be sold.
However, it's very difficult to see how such a restriction can be enforced in this particular case, given the fact that Thameslink services don't have reservations (not even counted places), so the service you are intended to take will not show up if you have a paper ticket (on E-Tickets and some M-Tickets it may).
Thisbeing the case, and the official instruction being that you should take an "appropriate service" for unreserved legs of the journey, I fail to see how an ordinary passenger would find anything other than the LNER service they arrive into Peterborough on, as the most appropriate service. They're all non-overtaken (for getting to King's Cross) and it's the most convenient option.
I would be highly surprised if any LNER staff would take umbrage to you travelling on an LNER service on an "LNER & connections" ticket, for the part where no service is specified!
Some suggest that you can only use unreservable services for unreserved portions of your journey but this is a restriction which is only communicated to ticketing websites. Customers are never told this and so it's quite clearly an unenforceable restriction.
You should, of course, in theory take your booked services only with an Advance, but this is contradicted by what National Rail says, and in any case how is someone to know what services they're booked on if they don't have the booking confirmation handy in front of them (or if they didn't even book the tickets themselves)?
You'll be fine continuing on the service in all likelihood.