So we are clear the smell is from the mains water not just when using the washing machine.
If that is the case based on washing clothes in the kitchen sink, do you notice the same problem when running a bath, if you do does it only occur when using hot\warm water, or is it obvious on cold water too?
Its a case of trying to identify the source, is it the boiler, kitchen sink\pipes, mains water feed etc.
Though the water has been tested this will look for harmful microbes, not identify a cause of smell, I assume none of your neighbours have the problem, and do you know if the water board checked if you have a water leak on the feed into your house?
I take it you havent noticed anything different about the taste of colour of the water if you put it in a glass, the fact the water doesnt seem to smell it sounds like it could be a reaction to the washing powder.
When washing in the sink are you just rinsing them out, or using washing powder, as I would try and give a few garments a good wash\rinse in the shower dry them, then proceed to wash them in the sink (without soap) and see if you notice the same smell. It could well be the clothes have absorbed some softener or soap which is reacting in some way.
A water softener will not fix this issue (in my view), this just strips minerals from the water and your neighbours arent having the same issue, I would try a different washing powder as it could be the washing powder your using has had its formula changed which is causing this issue.
The fact it doesnt make any difference which tap you use, and if its hot or cold water used I would suggest you flush the cold water system.
Do you have a combi boiler, or a header tank in the loft, it maybe there is something corroding in the tank.
I would be tempted to take the clothes and see if a neighbour, friends or relative can wash them to remove the smell, then wash them again in the sink with soap and see if the odour comes back, as it may well be the issue was something in the washer caused this and its taken a while for it to fade from the clothes.
I just wanted to follow up and check if you required any further assistance, and request you rate the service received so I may resolve this case.
No problem appreciate if you can rate the service received.
I await to hear back how you get on.