The 30-year-old midfielder has been a regular in Bury's starting lineup this season, making 26 League One appearances for the relegation-threatened Shakers.
However, he has now ended his five-year spell at Gigg Lane, and manager Neal Ardley admitted that he has had a long-standing interest in the former Crystal Palace man.
"Tom is a player we've liked for a number of years and we looked at maybe bringing him in back in November. We tried early in January to make enquiries, but it did not look like being something we could do. We then looked towards the summer for maybe a chance of getting him then if he did not sign a new deal," Ardley told the club's official website.
"We would have looked to put something in short-term until then, but quite late on in the window it started looking we might have half a chance. It took a lot of hard work yesterday, but we got there and I'm absolutely delighted we've got him. Tom has often played out of position this season at centre-back, right back, or right midfield, but his best position is as a central midfielder.
"He can play a holding role, advanced, or as a two. He is very powerful, has good aerial ability, and a lot of his athletic stats are very high. We've done a lot of homework on him and he is a very good player. If you look at his pedigree, he signed for Stoke from Crystal Palace for over £1m.
"I think he is in his prime now. When I thought there may be a chance of getting Tom, I asked the board to put together whatever they could muster to try to get a deal over the line and they have come up trumps. The board and the club have done a great job."
Soares could make his debut for AFC Wimbledon against Sheffield United on Saturday.