The 21-year-old had been heavily linked with a move to Kenilworth Road back in January, but has opted to leave the Robins relegated to League Two last season, and head to the Dons.
Twine scored 14 goals and had nine assists for the Robins and Newport County, whom he spent time with on loan last term, as when asked if Town had made a bid during the previous transfer window, Hatters boss Nathan Jones said: "Of course we know of him because we have our eye on every league.
"We probably could tell you what certain players have for breakfast but in terms of an offer, no it is not.
"He is someone else’s player so we've got to be very careful about speaking about other players anyway.
"He is a young player we know a lot of him, but we have a lot of young players here and that is why we brought Kal Naismith in to supplement a lot of young players we have.
"Kal is only 28 by the way but we felt that's what we needed.
"Whether it’s needed at this point, a young player for example, is yet to be decided."
Following news of his move to the Dons, Twine told the club's official website: “I'm an MK Dons player and I couldn't be happier.
“There were a few clubs interested but after I spoke to the gaffer (Russell Martin), he sold it to me.
"From that day, I couldn't wait to sign and get it sorted.
“I wanted to come here and play football the way we do, and I want to bring lots of goals and assists.”
“This is a club that is only going in one direction and that's up. The stadium is state of the art and to think I'm going to be playing here every week – I just can't wait!”
Dons manager Martin added: “He was wanted by a number of clubs but we’ve been able to act quickly in bringing him into the club.
"He’s desperate to be here and I know he’s very excited to get going - we’re just as excited to have him here!”