I think your best bet would be to start a campaign, and get organised. So when something happened in a previous place I lived and the council sat complacently and did nothing, after allowing them a reasonable time to deal with the problem , I decided that the drip drip method would be best. Had a meeting of like minded people and we organised a rota between ourselves, so that at every council meeting there was someone there to bring the subject up. we took it in turns to write to the local paper frequently, visited councillors in their constituencies, wrote letters to surrounding councillors to find out what happened in their area etc etc. Some people dropped out of my group, but others added to it and after about 5 months , when it was seen that we had absolutely no intention of letting the subject go, something was eventually sorted. I am sure that the fact that local elections were coming up had nothing to do with, especially when I said in a letter to the paper that I hoped that they would get their ideas out to constituants quicker than they dealt with this problem, and was it because no one wanted the poison chalice and would hope to lose the election!! Once the problem was sorted, whenever something occured that affected many people such as very badly reset pavements after such as Zoom have been digging it up etc, I or a friend would go to the next big meeting of the council and of course besides asking what was to be done about the situation, we would also say " we hope that it is not going to take as long as the last time" The point of that was to make them aware that we would be repeating the situation and constantly bringing up the subject until it was dealt with. This does not always mean it is sorted a way we would like , but it is not allowed to be just shoved around and ignored.
So in your situation I suggest, if possible try and get some photos of rats etc and any other vermin. Take pictures and note the date on them so that you can show measured deterioration as the time goes on and be able to say they are making it worse by doing nothing. Then contact environmental health and ask them to look at the situation and tell you what the legal situation is regarding them doing nothing even though they are aware of the problem. Look amongst the people who are living very close to this property and see who is classed as vunerable - babies small children elderly ill patients etc. Help these people to get letters from their doctors, hospitals etc explaning how it is making their health worse etc etc. Keep making a note, so you might have a notebook and every friday evening take a picture on your phone, and put in comments regarding how something is worse, how rubbish is now being left there etc etc.All this will be useful ammunition, so that when the council say it is not so bad etc etc, you have the proof of how much worse it is. When they say they havent the money - and I do understand how difficult it is for councils at the moment, but you can show them that again a stitch in time saves nine, and if it is left as it is and you can show how much worse it has become in a set time then they would be saving themselves money by dealing with it NOW!! Good luck and get a group together, the bigger the better as some people will be bound to have to drop out for various reasons.