So, if this phrasal verb is passive, then we need to add 'from', if it's not - then we don't add any prepositions.
Is it possible to say: 'They will cut off electricity to you' or 'They will cut off your electricity' is the only possible variant?
If the context makes it clear, you could just say 'they were cut off [and couldn't boil a kettle]'; or else - with no contextual support, 'they had their electricity cut off' - this is a use of 'had sth done' that doesn't imply anything voluntary (like 'they had their kitchen refitted').
Thinks: that's interesting, books that teach 'have sth done' don't - in my rather limited experience - always cover this...
FYI - the ssame phrasal verb works with a range of resources (gas, electricity, their main source of <whatever>...) and in one case (inheritance) the resource is often implicit: 'His father cut him off without a penny, and he had to make his own way in life.'