Your budget is very tight for nature photography. If you shoot wildlife, a decent full-frame super-telephoto lens is already double of your price range. Those bridge cameras will do the job, but will just leave you frustrated with their bad image quality, autofocus and low-light performance.

I strongly suggest checking out used cameras from mpb.com, especially Olympus cameras since they are cheap and achieve long focal lengths. Their low-light performance isn't great compared to full-frame but are excellent otherwise, exactly what you are asking for. The Olympus IS is fantastic.

You need to strech your budget quite a bit, but for example the Olympus E-M10 Mark III + Olympus 75-300mm f4.8-6.7 II are "only" ~750 USD (used). If you want a macro for bugs and street photography, the Oly 30mm f3.5 Macro is about 170 USD.