Oddly, our little used downstairs bathroom is bigger than the 'main' one upstairs. I'd like to use a little of the spare space to make it more of a utility room by sticking the washing-machine and tumble dryer in there (stacked - it's not 'that' big). But we also want to keep the full bath etc. Is this possible? Does the socket for the washing machine and the appliance itself need to be a certain distance from the bath or something?

its been a while since i have done some sparkey work but you can have a socket in zone 3(3meters away from a water source) with RCD protection under the 17th might need a confirmation on that as god knows where my onsite guide has gone since i moved over to the dark side

A bathroom is a "special location" so any changes to the electrics will have to be carried out by a registered electrician.

They'll be able to advise you what can go where and how it can be done when you get them in to quote.

Although you are allowed a socket 3 meters from zone 1 really it's not designed for bathrooms but for when a shower is fitted in a bedroom.

As to special zone yes you can DIY but likely it will cost more to get the LABC to issue the completion certificate than it would cost to get whole job done by a registered electrician.

Since you ask the question I would think the only way to comply would be to make the bathroom smaller and divide it into a utility room and bathroom.