Searching the SE1 forums for someone to mend our washing machine didn't throw up any useful (working) numbers so I tried the local (Walworth Road) electrical shop - Lynne's - 7703 7027.
They do do repairs and gave me the number of Russell - 07973 305 794. I had a choice of two slots and chose the morning 10.00-13.00.
Russell arrived at midday. He said he works London-wide. However, he knows the local area sufficiently well to find us. (Our 'street' is not in A-Zs.) I was also surprised (and pleased) that he didn't ask for a parking permit. He is aware of the effect of water on wooden floors and avoided getting our kitchen floor wet. He explained everything very patiently - enough times for me to be able to write it down for future reference.
The charge was GBP50 - this was the inclusive call-out charge. Parts would have been extra. If we need to call him out again - to replace a part or the whole machine - and we still have the receipt we will be able to offset part of the money paid this time against subsequent work.
He will be the person I call should we have further problems. I didn't think to ask if he can repairs on other equipment (sorry) but Lynne's sells all sorts of white goods so they likely have an engineer for most things.