"Shove up" is two words, therefore it would be an informal British phrase, not word.
I would say it is more than informal - it's downright rude. If someone (other than a very good friend or a relative) simply said "Shove up" to me, when they wanted me to make room for them, I would not be happy. I would expect someone to say "Can you move over please?" Yes, this is a common request of that type in AmE.