The 23-year-old, who signed a three-year contract at Bramall Lane earlier this week, wasted no time agreeing terms after the two clubs agreed an undisclosed fee.
Taylor said: "It all happened very quickly this week. I got a phone call on Monday night, did the medical and the contract the next day and everything was done. Then it was on a plane and off to Hungary.
"It was all done very quickly, but as soon as I knew United were interested my mind was made up.
"With us being away and together as a team a lot of the time I'm sure it will help me settle in. It seems as though they're a great bunch of lads and I can't wait to get started."