OP I'm sorry to hear this.
However my gut reaction is that you shouldn't move, even though I understand why you would consider doing so.
Next point - is she still taking her medication? If so then...
At some point (as an adult) she made the decision to move to London.
If you indulge the expectation that you'll uproot your life at the behest of her insecurities (I use that word specifically as you posted her mental heath is better but she "finds it hard to be without you") you will set a pattern for the rest of not only her life, but also that of your younger child.
London might be the best place for her career but you have to enforce if she can't hack it then maybe she needs to rethink her options rather than expecting the rest of the family to re-align their goals/expectations around her.
Harsh? Yes, but do you think her future employers will pander to her in this way?
They won't - so she can cut it or not. If not then why delay the inevitable.
So options...
- She stays in London but you offer to visit more frequently and offer daily support via telephone/Skype etc.
- she reviews her career path and looks to re-locate to a local uni course with a career path closer to home.
I'm sorry if I sound harsh, but she's now an adult that needs to take control of her own mental heath.