It takes longer to get your NIE through the Embassy, as they will need to send forms through the post and wait for them to be returned by post, and your application will obviously take it’s place in the queue with all of their other business, so best to just apply in person at the Police Station nearest to where you live when you get there - they do give it to you immediately, but best to get there very early, as sometimes, they only deal with NIE and Residencia applications in the morning.
You should be able to get your electricity and water account setup for you initially be your sales or letting agent on the basis that you will let them have your NIE and Bank details when you have them, but they will probably need a payment up-front to cover the first month.
I’m surprised you have got as far as buying or renting somewhere without an NIE so far!
It’s worth asking them to do this for you, as you don’t want to be sitting in the dark and having to flush your loo with Perrier while they get round to connecting you 😊 Things like this sometimes take a long time in Spain, but sales and letting agents usually have a good relationship with the utility companies!