My simple answer would be "Sorry, but no".
Probably mine too, though it sounds like she was rude and therefore that would tend the answer that way anyway. A polite request with an explanation - "Sorry, do you mind if my child sits down, please? We've been on our feet all day and we are both shattered", with gracious acceptance of "Sorry, I'm afraid I've been on my feet since 6am too, perhaps he could sit on your knee?",
might result in me giving up the seat depending upon how tired I was. Though I mostly stand rather than sit on the Tube anyway even if there are seats available, so if I'm sitting I'm probably knackered myself.
As ever, if you ask me politely and without entitlement[1] then I respond favourably in most cases unless there's a good reason not to. If you are rude you get nowt.
[1] Beyond what you are entitled to, of course. I normally sit in priority seats for the legroom, typically on the window side which has meant the number of times I have been turfed out is miniscule. But even so if you approach me rudely to sit there, I'll take my time gathering my belongings before moving. And similarly I won't move for a companion if they are impolite - they can stand. OTOH I have given up a paid-for exit row seat on a flight with no compensation and in significant discomfort for the flight because I was asked very nicely if a family could sit together and it solved a 3-way swap.