Ask UK

Flyers in the neighborhood, posts on FB and Nextdoor, tell friends and coworkers and ask them to tell people.

The way I see it you have, broadly 2 routes you can go down. You can contact the RSPCA (calling your local branch if you can.) Explain your concerns and your evidence of previous behaviour. If there isn't a court order in place they will be limited with what they can do, but they are best placed to track the situation and seek such an order. It will largely be out of your hands what happens after that.

Alternatively you can avoid involving state power at this point, befriend your new neighbour and monitor the situation. If she can be spared social isolation, there may be opportunity for her to have a healthy lifestyle and properly look after her cats. You will also be in a better position to see any problems as they develop, and provide friendly advice and support rather than just calling in a big hammer.

I know the RSPCA probably won't do anything and I'm not keen on getting someone evicted either. Any ideas?

The only way to guarantee they'll do nothing is to not notify them. Like you said, who cares if the animals are against the rules, as long as they're cared for. On the other hand, if the renters already have a history of neglect (and I would argue abuse)? Their lease agreement is not your concern if you're acting in the best interests of innocent animals...