One of my cats recently came back home after going missing for a few days and now his tail and one of his back legs had to be amputated. He is feeling better now but I wanted to ask if I should still let him outside and if I do if he will still be able to climb and run properly.
I've seen plenty of animals with missing limbs playing and exploring outdoors without problems. Your situation is a bit different though. Your cat is a recent amputee.
Losing a limb is a serious traumatic event. It takes a while to re-learn how to get around without that limb. A cat's tail is important to their sense of balance, so for the next few months your cat will have an increased risk of falling, missing jumps, etc. I highly recommend keeping the cat indoors for the next few months, and possibly even close off stairs and other potentially dangerous areas of the house. Keep a close eye on them and monitor how well they're recovering. Make sure they have the confidence and skill to use their new body configuration safely and consistently before you start introducing them to additional hazards. When it's time to re-introduce the outdoors, you may want to start by using a harness and accompanying them.
Also, a cat's tail plays an important part in the body language used to communicate with other cats. Your cat will suddenly find themselves unable to communicate effectively with its peers, which can increase the chances of getting in a fight. This again takes time to adjust to, and you should make sure your cat is completely ready before putting them in a potentially risky situation.
Your cat has just suffered a severe injury from being outdoors. There is no way to look at this and not conclude that the area he roamed was already unsafe for him.
So the answer to the question of if it is safe for him to go outside now, missing a leg and tail cannot be anything other than "no".
You will have to use your best judgement as to whether he will get enough enjoyment from going outside that it is worth that risk, knowing that he is now more vulnerable than he was before (the extent to which his missing leg and tail make him vulnerable will doubtless reduce over time as he adapts, but will never disappear entirely).
You should also consider the effect that outdoor cats have on the local wildlife bearing in mind your location. If you live somewhere where street cats are common like much of the Eastern Mediterranean, the cats have become part of the current ecological balance and there are so many that one pet, provided they're spayed or neutered, is unlikely to have much effect on that balance; in most of the rest of the world, especially places like the US and Australia, free-roaming cats pose a major threat to local bird species.
You should also think of ways you could adapt your home to make it more enriching for him, allowing him to live a happy life without going outside. Investing in cat furniture (including plenty of scratching posts), toys he can play with on his own, and taking the time to regularly play with him can all help here.
You can also consider allowing him outside in a controlled way where he is not free-roaming. This can be done with catios (an outdoor space the cat cannot escape from, and which dangers cannot enter), or through harness-training after which you can take him outdoors in a way where you can ensure he does not encounter dangers.
Indoor cats, or those only allowed out in controlled ways (such as those outlined above), can live full and happy lives if provided enough enrichment. They are much less likely to suffer serious injuries, their diet can be controlled completely (making it much easier to manage their health as they age), and it is easier to monitor their health meaning illness is likely to be caught early when it is more easily treated. As such, their lifespans are much longer (typically 10-15 years for indoor cats, compared to 2-5 for outdoor cats).
With that all in mind, I would strongly advise you not to allow your cat to free-roam again, and to instead keep him indoors, only allowing him out in either a catio or on a harness.
I don't think this question can just be answered about safety, but also about the animal's wellbeing.
Our cat is 3-legged (he was when we got him), but he does have a tail, so that may make a big difference in getting about, as they obviously use tails for balance, and for communication. He was always an outdoors cat, and was in quite a rural setting when he got the injury, then he moved to us - we are in the suburbs (the injury has nothing to do why he moved). It is quite a busy road, but not a main one. There are a lot of cats in our neighbourhood.
He gets about just fine, climbs fences (6ft) and can even still catch a bird/mouse or two.
If a cat DOES want to go out and you keep it in, I'm not sure that is best for the cat's quality of life - it could be safer to keep it in, but it could be miserable. Unfortunately there may not be a generic "one size fits all" answer, you have to evaluate what is best for your individual cat.
One of our neighbours has a three legged cat, it can run, jump and climb just as well as our four legged ones can. I've tried to watch how it climbs the 6ft high fence, but it's too quick!
You may well find that he's less confident than before, at least to start with, and so might hang around closer to home than previously, but if he wants to go out and is used to doing so, then you should let him.