"The Timeless Children" is an episode which has been hated on quite a lot because of the plot twist. Before we start‚i'd like to say there are no pre-hartnell doctors‚only pre-hartnell incarnations. The Fugitive Doctor is an incarnation of the doctor and it's one of those incarnations where we dont know where the hell to put her on the timeline,like that of the Curator. There are many theories,but I'd say that the Season 6B theory is most plausible. In The War Games,we never see the second doctor regenerate,we only see faces of him fly around himself as he falls into the void. The second doctor could've had a regeneration like the tenths,with him not actually changing. This leads us to the second second doctor and also Season 6B with the multi doctor specials. The second doctor eventually gets caught and regenerates into the fugitive and is forced to work for the Division as a punishment.
The Fugitive Doctor is not pre-hartnell,only post-troughton and pre-pertwee. So now we've proved the pre-hartnell fugitive theory isn't true,let's try and fix the actual problem itself:The Timeless Child.
How did the eleventh get a new set of regenerations if he had infinite? Because he didn't. The Timeless Child was used to give Time Lords regeneration abilities,but they only had 12 regenerations and 13 bodies. Why would The Timeless Child not also get 12 regenerations? The first face of morbius seen in their debut serial is eventually found and turned into the first doctor. The first doctor is very young at first but eventually has a grandchild called Susan. He takes on the alias "The Doctor",making him the first doctor. He got limited to 12 regenerations (those being him,troughton,pertwee,baker,davison,baker,mccoy,mcgann,eccleston,tennant and smith) and his memory was wiped.
This was until the Thirteenth Doctor (with a new ongoing regeneration cycle,with the only ones apart of it currently known being Capaldi,Whittaker,Tennant & Gatwa) re-met the Master and he trapped her in the Matrix,leading to her learning a lot of things about herself including The Timeless Child & The Faces Of Morbius (the fourth doctor might've had a slight memory of the faces of morbius as he exclaims "no not that far back" or something like that).