If Queen Margaret and Jasper Tudor had linked up and Edward IV had learned about it then he would have stopped his pursuit of the Lancastrian Queen and gathered his strength as he only had about 3,000 men with him (the Lancastrians had between 5,000-6,000 at Tewkesbury). Edward IV then would have called his supports in Wales to rally to his banner as well as called for men from western England before going after the Lancastrians. The problem for the Lancastrian forces is that Jasper Tudor most likely wouldn't have been Margaret's overall field commander, it most likely would have been the Duke of Somerset though Jasper would have been right under him as one of Somerset's "division" commanders. In that case, the advantage goes to Edward IV...now if something were to occur in which Jasper Tudor is selected by Margaret to be her overall field commander then it comes down to the terrain of the battlefield and how each man uses his forces.
If the Lancastrian link-up occurs without Edward IV knowledge and he stumbles on the combined force, Edward IV would be in a hazardous position. At Barnet, Edward IV had the smaller force (by how much is a matter of debate between the sources) but he destroyed the Lancastrians nonetheless. Now with just 3,000 men Edward IV would need to quickly find a defensible position or a quick retreat because if the Lancastrian get the surprise on him, Edward IV is looking at defeat or very quick retreat. While Edward IV is probably the best commander of the entire WotR period, the numerical disadvantage and sudden battle would have been too much for him.
Now if Edward IV were to lose against either a Jasper Tudor-led combined force or is surprised by the combined force, the surest way to get the Lancastrians some breathing room was for Edward IV to die in the battle. The House of York's heir would have been Edward IV's infant son Edward (V) then his three older sister, then his two uncles Clarence and Gloucester. The Yorkists would have had to retreat to Burgundy to await an opportunity to regain the throne. However maybe going with them is their captive Henry VI, who Edward IV captured before Barnet, unless some less than loyal Yorkist's release him and deliver him to the Lancastrians.
Let's just say that Henry VI is freed by Yorkist turncoats and unites with his victorious Queen and son. Henry VI will probably be King-in-name-only by now, I would assume Edward of Westminister would be declared by Henry VI (at his wife's suggestion) as co-regent and start ruling England with Margaret as his chief advisor. As Edward is married to Warwick's youngest daughter Anne Neville (OTL Richard III's Queen), the House of Lancaster could have several possible heirs within the next few years.
As for Henry Tudor, he would be restored to his inheritance of the Earl of Richmond. Considering he was a half-nephew to Henry VI and thus half-cousin to the future Edward V, he would have good marriage prospects considering his future inheritance via his mother. Could one of Tudor's son marry one of Edward of Westminister's daughters and the Tudor dynasty rule England through that connection? Possible, but the political situation would have to make that marriage advantageous to the Crown.