Or it can be that when he heats his cold water it kills bacteria that may be causing his issues, again this depends on how hot he he heated his cold water, did he allow it to go very hot and waited for it to get a bit cold to take bath without mixing further cold water into it, only he can confirm this, until he can tell us how he did and whether or not he mixed cold water into it, until then it is all speculation, I have warned OP that if he did go ahead and changed his supply to mains, there is no guarantee, and then he may well have to move his flat, or keep staying and seek specialist help on his condition, either way it is going to cost him some money or time.
So I hope everything goes OK for him, my Merry Christmas to all.
BTW, I was going to suggest to him, if he suffers from other medical conditions such as diabetes, and heavy smokers, as I am, It is not uncommon for diabetic people to suffer from rashes on lower part of legs and arms to come up with rashes after taking a hot water shower or bath, it is to do with blood vessels being constricted in diabetic patients and when they take hot water bath, the vessels dilate causing extra blood to flow under the skin and this sudden increase on blood rushing can cause an itch, and if one tries to relive this itch by scratching the skin can result in damage to the skin which may turn into blisters and deep rashes, if a person experiences such itch, it is best not to scratch that area but to rub the area gently to relieve the itch and apply itch calming lotions, but I know they don't always work instantly. i am diabetic so i suffer from a sudden urge to scratch lower part of my legs and top of the feet, there used to be times when i almost ripped my skin off the top of my feet! The itch is very strong, I don't get it any more and it was to do with allergy to my cats, I now take anti-histamine tablets and don't get this itch when taking a hot water shower. This is from my personal experience.
Indeed it was in my first post to OP that I asked him so many questions about his medical history then. I asked him if he suffered from any allergies and if he had seen his doctor and if he has prescribed him any anti-histamine tablets as those commonly used by hay fever sufferers. I now take one tablet every 3 days and all my itching is gone, occasionally I get intense itching on my head, and forehead and it comes up in blisters, when one of my cats chooses to sleep right on top of my head, again It goes away if I take one of those pills, or my doctor has also given me some steroid based scalp medication, that gives me instant relief. Thanks to these pills and scalp medication that i only have to use very rarely, my cats are not going anywhere yet, they are staying and they are allowed to sleep on my head, and I love it when they tickle my scalp! They may have gone out in filth, or killed a mice still who cares, I love my mogies. (Obviously I am not a medical doctor so i can't suggest him to start taking any anti-histamine pills, only his doctor can suggest or prescribe him those, even though they are available over the counter) But remember my condition erupted when taking hot water shower or when I took my shoes off when an intense itching would start on top of my feet, and this itch would then start spreading until the whole lower leg becomes itchy, spreads like fire! the cause was not related directly to hot water shower but rather allergy to my cats, may be OP lives in a flat where previously cats or other animals may have lived, and allergy can take time to develop, I was never allergic to cats before but now I am, so may be his carpets were not steam cleaned etc when he moved in 4 years ago, and there may still be some allergens and may be he also needs to look at having his carpets cleaned thoroughly.
So yes I agree that may be he should seek further help before spending on plumbing, may be if he speaks to his GP regarding a possible allergy and he may be able to prescribe some anti-histamine tablets that may help his condition better than ointments.