I would assume a local lockdown would limit what work can be carried out and as beauty was one of the last areas of business to re-open, I’d be surprised if you could continue to do mobile treatments in people’s homes.
However, for peace of mind, probably best to check with your local council, as I imagine the rules will vary in different areas.
From other areas that have been in a local lockdown, mobile hairdressers/beauty have still been able to work. Best to check with your local authority though.
It seems wrong to me to continue mobile but they don’t seem concerned about it.