It wont be pre-Hartnell, she has a police box . She also didnt recognise the sonic screwdriver which the 2nd doctor had. So IF she is really from 'our' tineline, it's likely she would be the 2nd doctor?
Still though, it makes a mockery of the whole 12 regen limit story line if that's this case.
The 12 regen story line has been broken and retconned all the way back to some of the early Master stories which were often around him trying to get additional regeneration either by stealing them from other time lords or via some Time Lord tech.
It's been a very long time since I read the stories/saw the relevant episodes but IIRC the Council could give additional regenerations to Time Lords who performed specific services to them (often I think to offset where a time lord lost a regeneration on a mission), and the old books basically said the ability to regenerate was not an inherent trait but engineered into the time lords and a result of one of their big bits of tech, between that and the whole regnerations as a reward/to replace those lost in service always gave the inference that the 12 regeneration limit was arbitrary and intended as much as a population control measure as anything else.
From memory the Lungbarrow story (book) and one of the old multiple Doctors stories explored this quite a bit, whilst multiple Master as the enemy episodes used the whole "wish to get more regenerations" thing.
Which just leaves the "why didn't the doctor remember.." but that's similar to the Doctor not remembering being the "War Doctor", or when the doctor went into hiding and sealed away his memories.
The Doctor Who timelines and abilities have always been rather fluid as there was never any "guide book" for the lore that was used (for example the first regeneration was because they wanted to keep the series going, but the actor was too ill) until very late in the day, by which time they'd already built up a nice set of conflicting bits of lore, most of which have never been dealt with
Instead they've built on it over the years, in much the same way that Star Trek has a load of conflicting information when you look into it, some of it dealt with by ignoring it, some by hand waving (IIRC the change in how Klingons looked at different times was brought up onscreen once and Worf said something like "We don't talk about it").
