Believe me, I've spent time figuring out how Patience fits into the Timeless Child backstory.
I keep going back to what I was thinking about yesterday -- that the TARDIS knows more about the Doctor in
An Unearthly Child than the Doctor does -- and I had an interesting insight, which meshes quite well with the TARDIS's story in
Time Lord Victorious.
What if, when the TARDIS reverted to the Police Box form and refused to change, she was trying to get the Doctor to remember who he was? She gets stuck as a Police Box, she kidnaps Ian and Barbara and doesn't take them home, and the Doctor, who had been a tourist, is forced by the TARDIS into situations where he has to be an
actor in the events, until over a series of adventures --
The Sensorites is commonly cited as the point where the first Doctor really starts acting Doctor-ish -- really starts to act like the person the TARDIS remembers he had been.
In other words, the Police Box is the TARDIS staging an Intervention for the Doctor.