I wouldn't say so. It's more likely than not, but The Timeless Children itself pointedly does not quite confirm that the Jo Martin Doctor was directly part of the pre-Hartnell Timeless Child's lives. The Thirteenth Doctor asks the Matrix projection of the Fugitive Doctor if that's the ticket, and she essentially goes “Search me, I only just got here”.
It remains a distinct possibility that the Fugitive Doctor is a Season 6B (or otherwise post-Hartnell) incarnation at a point when the Division tried to get their hooks back into their runaway agent. Is it the most likely possibility? No. But it's a possibility that is very consciously left open by the episode, and we should respect that.