Permanently! what like windows and doors?
Curtains. Can be bought from tesco and don't need to be installed by a competent person.
Same goes for washing machines.
I assume the nest isn't plug 'n Play. and something actually need wiring into the boiler/Control panel.
Something which you'll need a level of competence to do.
My Nan could disconnect a washing machine. Its a plug, a tap and a connector which should be hand tight.
Oh and she regularly removes her curtains to wash them.
Just stipulate on your law society inventory form that you give to your solicitor that that specific fixture you will be taking with you - no drama.
This.You can take/leave whatever you want as you list what's included on forms when you sort your contract for purchase/sale.

This.
Take it with you, make sure its clear on form, job done. When we move from this property I'll be taking a £1000 induction hob with me... Why? Because I'm not giving that in price for free!
You could also just extend them the offer of leaving the nest in on the form with a price you think is fair.
Personally I wouldn't be impressed if I moved into my new house and found the previous owner had swapped out the boiler controls!
Are you going to let the potential buyer know this before they submit an offer.
This is common courtesy, yes be upfront and say what you're leaving vs taking. Wouldn't waste someones time by hiding the fact I'm taking it.