Letting agent wading in….
Firstly, a standard assured short hold tenancy agreement should include a clause standing the tenant cannot run a business from the house.
If your tenant wanted to run a business from home they should have sought consent from you so that you can check the position with the mortgage provider and insurer. Also bear in mind that depending on the percentage usage of the property for the business this can affect rates.
Working from the home and running a business from home are two different things in the eyes of the council and HMRC.
As for the neighbour. I have plenty of neighbours that hate that the house next door is rented. It’s a snobbery thing and it grinds my gears! My grandfather owns a property in a cul de sac if busybodies who are totally the same about the tenants in his.
You need to handle this exactly as we do when it comes to this type of thing. Ask the neighbour to go and speak to the tenant directly. This is not your place to get involved (other than assuring yourself of the above matters). What would they do if the people were the homeowners and not tenants? People who cannot go and talk to their neighbour directly are the ones with the issue and are simply looking to cause trouble for people because they perceive that they can as they don’t ‘own’ the home they live in, it’s preposterous.
I literally had a neighbour the other day complain a bunch was still on the path outside the tenants house. Like, seriously, knock on their door if it’s really causing you such upset (it’s not a way that person would have walked and would not have affected them in any way the bin being left out). They then went on to say the people must live like pigs (they don’t, they are clean, tidy and lovely). I used to live in the house next door and I literally said to my husband ‘who knew we used to live amongst such obnoxious people’