Basic health care
Pigs are prone to all sorts of minor and major ailments from cuts to pneumonia. Spend time with them every day and use this time to check them over, looking for any physical or behavioural changes such as loss of appetite, excessive thirst, lethargy, vomiting, problems moving or walking, skin, hoof, mouth, ear problems.
Make sure you worm them regularly, usually every 6 months.
Regularly vaccinate against Erysipelas. Without rapid treatment this disease can kill pigs or lead to long-term issues like heart failure. It is very common as the bacteria can be picked up from soil infected by birds and rodents, and it can be passed from pig to pig. Speak to your vet about what's required. If you choose not to vaccinate, make yourself familiar with the first signs of the disease so you can act quickly. Information here: http://apha.defra.gov.uk/documents/surveillance/diseases/disease-smallholder-pet-pig.pdf
In Summer especially, watch for heat exhaustion and sunburn. A pig's temperature should be 38.6-38.8C. Keep a thermometer handy in case you need to check for fevers. If you suspect heat exhaustion, call the vet and then cool the pig down by sponging cold water over them and spraying the air around.
Check out which plants and household products can cause poisoning and make sure you keep any of these well away from your pig because as they will eat most things, they can easily get themselves into trouble, including swiping medications out of your bag!
Some diseases are notifiable to Defra. Make sure you know which ones you need to look out for, for example African Swine Fever (caused by infected flies, ticks and contact with infected pigs) is a big concern at the moment so you should take the time to read up on it on the government website. Pigs don't have to be in contact with each other to pass these diseases on so just because you have only a couple of pet pigs it doesn't mean there is no risk.
Key is to get to know what is normal for your pig and what isn't. Have a basic first aid kit to hand with antiseptic spray, thermometer, etc. Find a good pig vet and when you need them try to learn as much from them as you can that you can do yourself. Unfortunately pig vets really do vary widely in their knowledge and especially in their care and compassion, probably because most of their time is spent dealing with pigs that will be going for slaughter. So don't be afraid to try different vets if you don't feel the first one has the best interests of your pig at heart. At the least you want a vet who cares about your pig. Finally, for both smaller and larger pigs, there is an excellent, very well managed Facebook group called Mini Pig Health Management. And lots of useful advice can be found at: https://www.minipiginfo.com/mini-pig-health.html (please note this is an American site so some things may not be applicable in the UK).