This word order in English is unusual. "Come here quickly" is more usual, or "Quickly! come here"
I agree that "Come here quickly" sounds more natural.
I'd go further and say that we never say "Quickly come here". The only possible similar form is "Quickly, come here!" and even that sounds weird.
Наш кіт мушують зробити щось милий! Trying to say "Our cat must be doing something cute!" Not sure if I got it right.
Наш кіт, напевне, робить щось миле.
"Скоро ідіть сюди!" Is also another way of saying: "Quickly come here!"