Amber flashing lights are allowed on quite a large range of vehicles.
"amber light from a warning beacon fitted to–
(i)a road clearance vehicle;
(ii)a vehicle constructed or adapted for the purpose of collecting refuse;
(iii)a breakdown vehicle;
(iv)a vehicle having a maximum speed not exceeding 25 mph or any trailer drawn by such a vehicle;
(v)a vehicle having an overall width (including any load) exceeding 2.9 m;
(vi)a vehicle used for the purposes of testing, maintaining, improving, cleansing or watering roads or for any purpose incidental to any such use;
(vii)a vehicle used for the purpose of inspecting, cleansing, maintaining, adjusting, renewing or installing any apparatus which is in, on, under or over a road, or for any purpose incidental to any such use;
(viii)a vehicle used for or in connection with any purpose for which it is authorised to be used on roads by an order under section 44 of the Act;
[
F5(ix)a vehicle used for escort purposes;]
(x)a vehicle used by the Commissioners of Customs and Excise for the purpose of testing fuels;
(xi)a vehicle used for the purpose of surveying;
(xii)a vehicle used for the removal or immobilisation of vehicles in exercise of a statutory power or duty;"
The mobile tyre fitter will class as a breakdown vehicle for that purpose. Not sure about the mobility scooter vehicle.