I think it depends on the exact circumstances (which, since the story is still in draft, you may be able to tweak?).
The most important thing is whether there is a birth certificate (or other record) of a son born to Cora while she was married to Robert. If there is, then the matter is fairly straightforward–and if there isn’t, it may be impossible.
If there is a record of a legitimate son, then they don’t have to prove that Thomas is the heir. They have to establish that he’s Robert Junior (or whatever name the birth was registered under), and the heirdom comes automatically. I’m pretty sure all it takes is the family making some kind of announcement, “Good news, everyone! We’ve found Robert Junior” (or whatever name), and then everything proceeds as if Robert Junior had had a more typical upbringing.
You could handwave the whole thing with Robert (or someone) saying something about “have Murray take care of the details” and move on, as long as nobody who has standing to do so (like Matthew, who’d be disinherited) objects publicly.
If you want to portray a formal, legal process, on the other hand, you could introduce a surmountable level of complication by saying that Robert Junior had been previously declared dead. In that case, the declaration would have to be overturned. I haven’t found any period-specific information about how that process would have worked, but a court of law would have been involved; you could do a scene where they lay out the evidence before a judge. Again, if the Crawleys are united in saying “Yes, we’re sure this is Robert Junior,” and nobody (with standing to do so) is objecting, it could be pretty pro-forma.
Another complication would be if Robert Junior was not only declared dead, but buried. Like, say, a few months after he was kidnapped, remains were found of an infant the right age and sex. (If the remains were too deteriorated for visual identification, they wouldn’t have had much to fall back on, forensically speaking. Not just no DNA, but no fingerprints/footprints or even blood typing.)
In that case, there would have been a coroner’s inquest, which would have to be reopened in order to have another go at identifying the deceased. (Even if the result was just “unknown male infant.”) That would probably involve a review of the evidence that led the inquest to conclude that the deceased infant was Robert Junior in the first place, as well as the new evidence that it wasn’t (which is to say, the evidence that Robert Junior is still alive and standing right in front of them).
Again, all of these presuppose that the heir–Robert Junior, Robert and Cora’s son–legally existed at some point. (England started requiring legal registration of births in 1837, so that isn’t too heavy a lift. Not everyone actually bothered, but it would be very unusual for a baby born to a married woman of high status not to have a birth certificate.)
If, however, the birth was never registered? In that case, I’m, like, 95% sure that the answer to the question “How do they prove Thomas is the heir?” is they can’t. Inheritance is as much about legal ties as blood ones, and the legal system does not have a mechanism for an adult son appearing out of nowhere*
(*Yes, yes, there’s the Earl of Findlater. But even if you say that Cora gave birth to Thomas before the wedding, and left the birth unregistered for that reason, you’d also have to move Downton Abbey to Scotland. Legitimation by subsequent marriage doesn’t apply to English titles.)
Anyway, or narrative purposes, if it’s necessary to say that there was no record of the birth, you’re pretty much stuck inventing a process, and have the characters hang a lampshade on the fact that this is an unprecedented situation so they have to improvise.
Just spitballing, I’d probably have the process focus on retroactively registering the birth–that is, establishing parentage and legitimacy–rather than anything directly related to him being the heir, for two reasons. One, the whole point of primogeniture and entail is that you don’t get to pick who your heir is. Two, I’m finding some information about a process for late registration of a birth. It isn’t clear whether it was possible to do so in the period in question–that link references a 1965 revision to the Registration of Births Act–but it’s at least something to go on. (For instance, the relevant office is called the General Registry Office, and the official who’d need to approve anything nonstandard is the General Registrar.)
Anyway–interesting question; I hope this helps!
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