Maius Intra Qua Extra

Now that's an interesting question! I wonder, for example, what Martha would think if, 25 years in the future, she finds that her (and Mickey's) son or daughter began traveling with the Doctor? A lot would depend on if the parent(s) had already told their children about their travels. I might have to think about that for a future story.

About the canon mothers, though...

I'd think that of the three, Francine would not like the Doctor no matter in what context she met him. She's very status-conscious and his informal, flippant attitude would turn her off right quick. I think both Jackie and Sylvia would notice that he's strange, but wouldn't dislike him. Then, if they were to find out that he's an alien that travels around and saves civilizations... Well, I think Jackie would first be horrified that he's an alien, but then she'd 180-flip to hero-worship. Sylvia would be much more level-headed and appreciate him.

THEN when their daughters meet him and want to travel with him. Well, nothing's going to change Francine's mind. After the Year That Never Was, while she appreciated that he saved the world, I do not believe that she still considered him safe for Martha to be around. Jackie wouldn't approve of Rose traveling with him, because she's extremely protective, but she wouldn't be able to stop her.

Of the three, I think that Sylvia would be fine with Donna going off with him. I think that all of the objections she had in the show to Donna going off with him stemmed from her not trusting him. If Sylvia had known the Doctor beforehand and understood him, I think she would have approved of Donna going with him, while always still being concerned for her safety. After all, all Sylvia ever wanted was for Donna to work hard and realize her potential, and the Doctor would help her do that.