The 21-year-old rejected the opportunity to remain in Shropshire, and after the two clubs agreed a compensation package, Saddlers boss Dean Smith is delighted to have strengthened his attack with the signing of the former Wales Under-21 international.
The 43-year-old is quoted by the Express & Star as saying: "We are delighted that we have finally been able to secure Tom's services. We know all about him and the qualities that he will bring to us.
"He's a player that has always caught my eye whenever he has played against us at various levels or I have watched him."
Bradshaw, who missed part of last season with a knee injury, recorded seven goals in 30 appearances during the campaign.