The most important advice is not to put yourself at risk! Injured foxes, badgers, deer, mink and weasels can be dangerous. Just cutting wire or removing other trapping situations and releasing the animal may not be the right thing to do as the dangerous effects of long term restricted blood supply may not show immediately and although the casualty may escape it could be carrying a life threatening injury. The same applies to an animal stuck in a drain or under a building as it may be dehydrated or hypothermic. Please call for advice and help.
Baby mammals are cute and the temptation to pick them up is overwhelming. However PLEASE take time to be sure this is the best thing to do. If the baby is at risk, from predators, traffic or its environment then maybe there is no choice but Mum may not be far away and watching for you to move before collecting her little one. She may have been in the process of moving her family or in the case of deer the baby knows to lie very still for many hours while mum is away feeding. If you can, leave well alone and return a couple of hours later to check on the situation. If there is no choice but to get involved then handle the casualty as little as possible using a soft cloth and/or wearing gloves. Put it into a suitable box lined with torn up paper (not hay please) avoid checking on how it is doing, do not offer food or water and make sure a full assessment is carried out by an appropriate person.
With the dangers involved in handling the larger mammals it may be best to call for help rather than risking getting bitten, butted or kicked. Stay with the casualty if at all possible while you are waiting for help or at least mark where it is as clearly as possible with sticks, something tied to the fence or arrows in the mud. The injured animal may try and move when you have gone and there is nothing worse than failing to find a casualty knowing it is seriously injured.
Hedgehogs are different. Very often if they are out and about in daylight then something is wrong. Occasionally it may be that a mother hedgehog is foraging for her family but they are nocturnal creatures and avoid daylight as much as possible. Wait watch and listen. Coughing and wheezing are bad signs and urgent help may be needed. Failing to curl up or run away as you approach are signs the animal could be very sick or injured as are signs of diarrhoea or vomiting. If there are no signs of injury and the ‘hog is tightly curled then move it to a shaded location, put wet or dry cat or dog food near by and watch. If as the light fades there is no movement or attempt to feed then something is wrong. A safe box lined with paper is the best transportation to a full assessment or please phone for advice.
There is the problem of fleas! Yes, hedgehogs usually have dozens which is unpleasant, but the fleas will not live on you or your pets, in fact should a flea bite you it will die as your blood is totally undigestible for it!!!