I am fascinated by "huge overgrowth encouraged rats and other vermin".
What other vermin is that?
We get rats coming through the garden (there is a Domino pizza at the bottom of the garden and they don't seem to be terribly careful about what they throw away or in what sort of container, so of course there are rats in the gardens all around them) but I am trying to think what other vermin would be particularly attracted by an overgrown garden. Um. Cockroaches? You won't as a rule get mice where there are rats, as far as I know. Ferrets, stoats and weasels? They seem a little improbable in an urban garden....
When we moved into this house, the entire back garden was a rather scruffy lawn, with a magnolia tree in one corner and a flowering cherry planted about two foot from the foundations of the house. After a bit we got fed up with it, had the lawn removed (some blokes came and rolled it up and took it away in a pick-up) and rotivated the area, then bought seeds for a mix of nineteen flower species and six grasses and threw the seeds on the bare ground and left it alone unwalked on for six weeks.
We also bought a scythe.
Each year since has had one flower dominating: one year we had ox-eye daisies up to our waists in a solid mass, for instance. And this year for the first time we had cowslips in the spring. It gets scythed down twice a year to give the next lot a chance to grow, once in June and once in late September, and otherwise it is left to its own devices and to the birds, bees and butterflies.
The only real problem we have is a gurt bramble which has come in from the immaculately-manicured garden next door, where it was growing between their shed and our fence next to the magnolia, and which has got a hold inside the magnolia where we didn't see it. I am not sure what to do about it, to be honest; getting rid of it is going to be a hellish labour, because just unwinding it from the tree before it strangles it will be arduous and probably painful, and the root is in someone else's garden under her shed, where she can't get at it. Someone upthread said cut it back as far as you can and then put herbicide inside a bag round the top of the stems that are left, I think, and that's the first real hope I have heard about it all year, so thank you.