I've enjoyed taking pictures of wildlife on my phone for a while, especially birds, and would like to get a camera to get better quality images. I would possibly like to try some astrophotography too. I want to make sure I'm not wasting my money so would like to know what features to look for and what isn't as relevant. Also, any camera recommendations would be appreciated too! My budget is on the lower side, maybe £200-£300.
I would start with a point and shoot with as long an effective focal length lens as you can afford. Wildlife is generally far away and even 250-300mm 35mm-equivalent focal length can be very frustrating. It takes practice to find the subject and hold the camera steady. Image stabilization and/or a tripod is required. The lenses are not the sharpest because of the long zoom range they have and the images are a bit noisy due to the small sensor, but the photos are still satisfying, especially in good light. Shorebirds are a good subject as they are often out in good sunlight and some of them are reasonably large.
I'm going to throw my hat into the ring with - a crop-frame DSLR such as the Nikon D5xxx or cheaper D3xxx series. eBay - used D3300 £160, used D5500 £280
I would marry that with a long-throw zoom. These are not the best lenses you can buy, but they're a good start until you get enough experience to know what more specialist lenses you may want/need later.
My walkabout zoom which I use for everything from landscapes to small wildlife closeups is the Nikon 18-300mm - but at £400 second hand, this is right out of your price range. Sigma also do an 18-300, slightly cheaper, which I've never tried.
Below that are such as the Tamron 70-300 which are only about £100 new & can be picked up for under £30 on eBay. The trouble with these is they have no 'anti-shake' and they are slow to focus. They are, on the other hand dirt cheap & will fit your budget. I started with one of these, and it was fine for a beginner. You just need a bit more patience with it.
The advantage of a zoom, for a beginner, is you can find your subject whilst zoomed out, then zoom in whilst keeping it in frame.
It's not too easy when you're starting out to aim a fully-zoomed lens to find some small creature off in the distance.
A 300mm is just adequate for things that will stay still & not be bothered by you getting close - a bee on a flower etc which will just do its own thing & won't care about you - or a squirrel. You can walk to within 10ft of a squirrel, but any closer & it will get skittish - or if you have a pocketful of nuts, it will be at your feet where you can't take a good picture.*
Birds smaller than magpies you won't get close enough on most occasions to properly fill the frame, so you have to crop afterwards, which loses definition.
*Fun with squirrels - if you kneel 10 ft or so in front of a squirrel [with no other people/distractions around] facing it and looking directly at it, it will come towards you. Have your camera in both hands ready to go. As it approaches, lift your camera with both hands quite quickly to your face.
The squirrel will adopt its 'sit up & beg' pose. You can repeat this as often as you like. Each time it does it, it will remain absolutely still for several seconds. Perfect photo opportunity.
[Set your camera to silent mode, the focus beep will scare them, but the shutter release won't.]
For your budget, the best you can do is a cheap used crop sensor DSLR plus a 55-250 zoom lens (on Canon 55-250 is discontinued so you may need to choose a used one). It's a rather low budget for photographing birds, and to get that for 300 GBP you will need to search hard for used bargains as 300 GBP is a rather tight budget even for the cheapest possible gear. You will soon find that your gear is a limitation, though. It's however far, far better than even the best phone camera.
For birds, my main Canon recommendation (some Nikon or Sony user could recommend something for their systems as I don't know anything about those systems) would be either a 70D/80D/90D series crop DSLR with 400mm f/5.6 (about 2000 GBP; the 400/5.6 is discontinued so only a used one is an option) or R5/R6 full frame mirrorless with 800mm f/11 (about 4000-6000 GBP). However, both of those are way beyond your budget range, and the fixed zoom lens works only for birds -- for larger animals, you would need a 55-250 or 70-300 for crop camera or RF 100-400 or 100-500 for full frame mirrorless.
You will quite quickly observe that the 55-250 isn't terribly sharp and the focal length leaves a lot to be desired.
Here are two photographs of moon. First is 55-250mm with crop camera, the second is 400mm f/5.6 with full frame camera. The focal lengths are equivalent (250mm crop is same as 400mm full frame) and the cameras had about the same megapixel count (24 vs 26.2). So the only variable is lens quality. You will see that the 55-250 isn't a terribly good lens.
Also 250mm on crop or 400mm on full frame is bit short for birds. That's why I recommend 400mm on crop and 800mm on full frame.
For astrophotography (apart from moon photography which you can do with wildlife gear, because the moon is very bright because it's illuminated by the sun), you will need a wider fast lens, the wildlife lens won't work. Or if you use a long lens to get not wide-angle Milky Way photographs but rather photographs of an individual far-away galaxy, you need either a tracker or maybe you could try image stacking. The problem with far-away galaxies is that Earth is continuously rotating and they are dim. Collecting much light with long exposures would just display the fact that Earth is rotating. With wide-angle fast primes, Earth's rotation is less of a problem because of the wide angle (so the exposure time for which Earth's rotation is a problem is longer) and because a fast prime doesn't require too long exposure times.
Wildlife and a camera are wildlife photography’s only must haves. Everything else is a matter of personal preference.
While it is probably a good idea to use a long lens to photograph polar bears, a normal or wide lens is probably better for framing a flock of seagulls against moonset over pounding surf. A trail camera would provide a whole different set of potentials.
A pair of sturdy shoes, a tripod, and time away from the office are also useful.
So is patience. Moose Peterson describes his one true skill as the ability to sit and wait for ten hours for the one moment when the animal pops up briefly from its burrow.
Which is all to say that since you are making wildlife pictures with your phone already, it won’t matter what gear you buy.
So long as the new gear doesn’t get in your way. If it does, ruthlessly sell it on eBay and try something else. Finding what works for you will be trial and error.