Is meeting the entry requirements enough to justify applying for an MSc?
Previously people submitted three applications, one certain, one stretch, and one intermediate. To use the work experience to your advantage, try to find programs that encourage people with relevant work experience to apply.

In the US I'd advise you not to mention the mental health diagnosis at this stage.  However, I don't know if that carries over to the UK.

Even without mentioning specifics, it is still possible to write a short accompanying statement about the poor grades early on, something along the lines of "facing a challenging family situation during years x and y" and pointing to recent spectacular successes as a counterbalance.

In general you should aim for a non-apologetic, self-confident tone, especially since you want to go into finance -- which I think is a field full of chutzpah.  If you are not accepted, you're not accepted; but no one will judge you harshly for having tried.

With regard to the money -- look, in the finance world, don't they talk a lot about "the cost of doing business"?  Two or three applications are not going to make much of a dent in your nest egg.

By the way, I wonder whether you would like to consider looking beyond your national borders?  Finance is such an international sort of field.

(I actually agree this should be closed but I had too much to say to fit into comments.)