Illegitimacy and inheritance

[JamieS]

A housemaid working stately home gets pregnant by a Lord and gives birth to a son. There's more to the story, but that's the gist. Here's my question, if the male decedents of that baby can get DNA proof that they are related to a specific family, do they have any claim to that family's peerage (in our case it would be a Viscount)? Especially if the baby was the second born son and the title currently is being held by the 3rd son's decedents. I know back in the day illegitimate children couldn't inherit anything, but did that change with the Family Reform Act? Is it possible for our grandfather to become heir?