I know the examples I gave might not seem like much to people it’s very very difficult to convey just what a complex character she is
Don't worry op, I get it. In fact, without wishing to be over-dramatic, your post sent chills down my spine ; when you describe your mother, it's as if you are describing my late mil too. The complex character: very wilful, manipulative, intelligent, independent, impulsive and extremely domineering on the one hand, and almost toddler-like, impractical, needy, contrary, and completely lacking in common sense on the other. Competent , decisive and dynamic over something that interests and affects her (world travel?) , and conveniently incompetent and scatter brained when it comes to something that feels too much like hard work (such as the harder aspects of looking after small children).
She is similar right down to the packing up of the house! Me and my DH have done this for my mil three times now (long story), and the flat where she last lived (now occupied half the time by another family member) looks like it is going to have to be sorted out by us again in the near future, and a storage unit. And I don't think I have recovered from the time spent cleaning and packing up their house in the uk when (similar to you) I kept coming downstairs to find that she had unpacked yet another box that we had packed the night before! The piece de resistance was when she unpacked the kitchen boxes to cook the removals men on the morning of the move, a full English breakfast, leaving greasy saucepans, grease spattered work services and grill, plates and cutlery, which had all been cleaned and packed up the day before . An "essentially benign" act one may say for the removal men. But it was also deliberately obstructive and disruptive given that my DH had travelled from abroad and given up days of unpaid holiday to help her move and not only did she not lift a finger to help ( she alternated between becoming hysterical or sitting in a chair criticising everything was being done incorrectly) she put obstacle after obstacle in our path, even though the move had been initiated by her, as she wanted to go and live in a different sunnier country!
I feel every sympathy for you. Unless you have experienced having someone like this in your close family, and even worse have had to live with them for long periods of time, then you can't understand how hugely frustrating, utterly exhausting and demoralising the experience can be.
We thought that my mil was "essentially benign" too, and in her earlier years, she may have been. But looking back, my dh and I, like you and your dh I suspect, were too naieve, too dutiful, too willing to help and we did not put enough boundaries in place. Ultimately, we had to, when her erratic and selfish behaviour impacted directly on our DD, and I suggest you do the same before the situation deteriorates any further.
In fact I would suggest reading up on FOG - fear obligation and guilt (it is a thing) and then look at your mother's behaviour again with a more dispassionate eye.
Personally I wouldn't have accepted her money to help buy your house as it makes asking her to leave very difficult and makes you financially beholden. But had that not been the case, and having been through a similar experience, I would ask her to go without hesitation.
I'm afraid it is a classic manipulator's move to give you money for your own house and imply that they are happy to downsize to a small flat in London and live alone there, and then when the time comes , to claim that they don't have enough furnishings and they want to live with someone else. Did she mention needing to live with someone before the two new properties were acquired? ? I can hazard a guess that she didn't.
If she was independent and healthy enough to travel extensively recently, why is she not well enough to live in her studio flat now?
Op I am sure that what I am saying goes against the grain as you sound kind, sensible and well-meaning. But I am saying it as someone who has a close and loving family themselves and who understands the importance of close family relationships, and who (ordinarily) is not in the habit of suggesting people go "low contact" or "no contact" with their mothers! But that is precisely why it took me so long to work out that my mil was being manipulative as I had never encountered anyone like her before. You are so close to her, it is perhaps difficult to look at this situation dispassionately. Unless your mil has a mh issue or some sort of impending dementia, her behwviour is manipulative in the extreme. People who genuinely love one another are selfless, they try and make your life less difficult, even when it inconveniences them. They don't tey and frustrate you at every turn.
Most important of all, despite the love and obligation you feel for your mother, please do NOT let her actions and behaviour destroy your relationship with your DH and disrupt the precious early months and years with your baby son. 