Does using one offer to "pressure" another school actually work?
Simply say that you are writing to let them know that you have received an offer from institution X. If there is a deadline for you to answer the offer, tell them what it is. This may be helpful for the department chair to use in pressuring their administration or other involved parties to make a decision sooner than they would have otherwise - yes, people can often "hurry along" when the circumstances warrant it. It may also have a psychological effect of making you look more attractive and desirable, especially if institution X is a prestigious one. But let them be impressed by the news itself; keep the tone of the email neutral and informative and avoid coming across as smug or entitled-sounding.

As for what to expect, that seems beside the point. Just like you would update the places you've applied to if you had a paper accepted to Nature since that would make you a more attractive candidate, you want to let them know about a job offer from another institution. It may or may not have any positive effect, but it sure can't hurt.

If it matters (I don't think it does, but you asked), I'm in the US. Good luck!