1455 & 1459-61
England's loss of her French empire was a vital element in the background to the battle of St. Albans in 1455, but there was no direct involvement in the battle or its immediate aftermath. Similarily, there was little direct involvement between 1455 and 1461, although the French raid on Sandwich in 1457 shocked rival nobles into peace-talks. When conflict began from 1459, neither side wished to be seen, using foreign soldiers against their own countrymen. Charles VII of France did provide Margaret of Anjou (his niece) with a little diplomatic support and Burgundy may have paid for a small contingent of Yorkist troops at Towton, but that was all. Slightly more influential was the prescence of the papal legate, Francesco Coppini, with the Yorkists as they landed in Kent in 1460 and marched to London. Coppini had been sent to England by the Pope to get support for a crusade and decided, seemingly, on his own initiative, that York was more likely to provide this help than Henry VI. As a result, the papal banner flew over the Yorkists as they marched through Kent. Some locals may have confused Coppini with the Archbishop of Canterbury and thought that York had the backing of the Church and was therefore in the right.
After Towton, Louis XI of France provided Margaret with diplomatic and a little military aid but did no more. His priority was not defending Margaret but enduring France was not attacked by England. The danger for Louis was provoking Edward IV by giving too much aid to the Lancastrians. Scots' aid for Lancaster was of more practical help but was in Scotland's interests and not the result of 'taking sides' on behalf of Lancaster. The Scots' motive was winning the border castles at Berwick and Roxburgh. Thus, foreign aid helped to keep the Lancastrian cause alive but was unable to do more.
In 1470 France gave support to Warwick and Lancaster and in 1471 Burgundy helped Edward IV. Louis XI backed Warwick because he wanted to destablise Edward, who looked as if he might invade France. Importantly, Edward also looked to be in a weak position and ripe for defeat. Louis arranged the meeting in 1470 between Warwick and Margaret of Anjou that led to their alliance and provided money and ships to take Warwick and his supporters back to England. Louis got Henry VI back on the English throne, a king who was no threat to France.
Edward fled to Burgundy but Duke Charles wasn't his natural ally, even though he was his brother-in-law. Charles had family links to Lancaster, and was a close friend of the Duke of Somerset. Charles' priority was defending Burgundy against France and so he had tried to negotiate with Warwick and feared that Edward's arrival in Burgundy might push Warwick into an alliance with France. Nevertheless, Warwick had planned an alliance with the french.
Therefore, other people's decisions pushed Charles into helping Edward. Charles now knew that the only way he'd get English help to defend his lands was if Edward were back on the throne. He gave Edward £20,000, ships and facilities to prepare an invasion fleet. This Burgundian help created the springboard for his invasion and re-conquest of the kingdom.
James III of Scotland played no part. He was busy establishing himself on the throne and the Scots had control of Berwick, so there was no need to take advantage of English divisions.
When Richard became King, the Duke of Brittany first sought his help against France, but then, realising no aid was on the way, supported Henry Tudor's attempted invasion in October 1485 by providing him with ships. Afterwards, Brittany continued to provide refuge for Henry and his supporters until they fled to France to avoid being handed over to Richard.
In France, political divisions after the death of Louis XI in August 1483 eventually played into Henry's hands. The Regent of France was facing a combined threat from Burgundy, Brittany and England, all supporting rebel nobles. To break up the opposition alliance, the Regent supported Henry's invasion in 1485 as a way of destablising Richard. This was success but, ironically, by the time of Bosworth, events in France were going to Regent's way. If Henry had delayed his invasion by a month, there would have been no need for France to support him in toppling Richard.