Hopefully, someone from the U.K. will answer your question. My answer might bump your question back up to the top of the list.

In the U.S. older homes are grandfathered; nothing needs to be changed to meet modern code. I suspect your laws are similar as no one forced the former owner to rebuild the stairs before you moved in. Of course, if you rent, you would need to get permission from the landlord before changing or adding any structure.

While older homes are generally grandfathered, you can not, at present time, modify the stairs to be less code compliant than they already are.

I would guess that someone has either read the wrong part of the code, not reading far enough into the exceptions, or is confusing and mixing together wording from both the commercial and residential code.

Here is how the confusion works in the US. The commercial code always requires handrails on both sides, so it is logical to measure the clear distance between handrails.

The residential code gives you the option to leave off one of the handrails, and still allows you to add the second handrail when you get older, or develop a condition where two may be necessary such as the loss of the use of one arm. (Going up you would use the opposite rail as when going down) THEREFORE, it makes more sense to stipulate the wall to wall width of the stairs to accommodate both one and two handrails situations.

Otherwise, the code would have built-in age and handicapped discrimination issues where you would either be required to rebuild your house, or move out at 75. :-/

...and you apearantly got the "between handrails" (1.15b) for "other than dwellings" mixed up/into part 1.16 ... which is actually wall to wall ;)