Though I think both Edward and Thomas are possible names, I agree with isabella that "John" is the most likely name for the boy, as Henry V seemed to be closest with his brother John in OTL, and their grandfather was also named John, so I'm just going to refer to ATL Henry V's second son as "John" hereafter for simplicity's sake.
I don't imagine that John's existence meaningfully changes the course of the Lancastrian War during the 20s or early- to mid-30s, as he would obviously be a boy at this time, but considering that the English were still fighting for the establishment of the dual monarchy at the time of Henry VI's coronation as Henry II of France, I would guess that John is created a major French lord in addition to a major English lord -- very likely duke of Clarence in England and perhaps duke of Normandy in France, or duke of Anjou and count of Maine after the death of Bedford. This gives John a vested interest in the conquest of France.
Assuming ATL Henry VI is as extraordinarily pious as he was in OTL, I would guess that still comes under the influence of his cardinal great uncle early in his life. This allows the Beaufort party to emerge as the dominant faction in ATL 1430s, as it did in OTL.
Richard of York is almost certainly still governor of France after the death of Bedford in ATL 1430s, as John is too young for the role at this time and York is effectively the only candidate acceptable to both Cardinal Beaufort and Gloucester. Perhaps John is awarded the governorship of France in 1440 instead of York getting a second term, though. John would be 18 by this time, so he could in theory take such a command, though more likely he'd get the title and the salary and York would get a separate title and salary (perhaps lord lieutenant of Normandy?) and the two would work alongside one another, allowing John to learn administration and diplomacy from York and military command and strategy from Talbot.
Assuming John is militarily and politically competent -- he needn't be a Bedford-level genius, simply competent -- then he may recognize the corruption of the court party as he approached his majority (i.e., late-1430s/early-1440s). He would certainly be aware of the court party's incompetence, if not their corruption, as he fought in France to take control of whatever lands he'd been promised.
John being in France in the early-40s may butterfly away Somerset's doomed Gascon campaign, as John may be able to convince Henry VI to send the men and money to Normandy instead, whereas York tried and failed to convince Henry VI to do this in OTL. This could significantly change the course of the war, as Somerset's campaign was not only a military failure, but effectively bankrupted the crown and upended York's negotiations for a possible alliance with Brittany and Alencon.
If John and York get the money and the 8,000 men for Normandy that OTL Somerset got for Gascony, then the influx of men and money may convince Brittany and Alencon to make the jump and join an English alliance. John and York can secure the borders of Normandy and Maine and mount a new offensive toward Paris, perhaps retaking Pontoise. This is great news on the military front, but the campaign likely bankrupts the English crown just as Somerset's OTL campaign did. The English need to sue for peace in the mid-40s. The French agree to open negotiations, as the count of Armagnac threatens rebellion in the south.
The English are in a much stronger negotiating position in ATL after John and York's successes and the new alliance with Brittany and Alencon, but likely not strong enough to get Charles VII to agree to cede Normandy to England in full sovereignty (their OTL demand for a permanent peace at this time). Still, Suffolk could secure the truce and perhaps the marriage to Margaret of Anjou -- likely a major goal of talks if, as in OTL, Henry VI becomes obsessed with the girl after seeing her portrait -- without ceding Maine. Perhaps Charles VII -- facing the Anglo-Breton-Alencon alliance in the north and a possible Armagnac rebellion in the south -- is in a weak enough position that he seeks a longer truce than in OTL and perhaps even agrees to pay a dowry for the girl so she does not go to England penniless.
A longer truce, as well as the death of Cardinal Beaufort and Gloucester in 1447, gives John a chance to return to England and attempt to sort out the crown's finances. I don't think there would be any major conflict between John and Suffolk in the late-40s. Suffolk was resisting the court party toward the end of the cardinal's life, even if Suffolk was often the man that the king tasked with executing Beaufort's policies. Suffolk may emerge as a reformer if there is a figure like John who has the king's ear and with whom he can ally against the Beauforts. The restoration of basic governance and also having a major figure like John on the international marriage market could put England in a position to defend Normandy and Gascony for years longer than they did in OTL, though ultimately I think French victory is inevitable unless ATL Charles the Bold comes to power in Burgundy and restores the Anglo-Burgundian alliance. And could the English cling to power in Normandy all the way until Philip the Good's death in 1467? Seems like a stretch.
If John and York do
not get the money and the 8,000 men for Normandy, and ATL Somerset launches his doomed campaign into Gascony, then England's position in Normandy likely continues to slowly fail and I suspect John's experience in France radicalizes him against the court party, as it did York and others in OTL. Gloucester is politically isolated and powerless by the early-40s, and so John has no natural ally besides York and other Normandy veterans to oppose the court faction. (Even lords who later emerge as reformers, like Buckingham, were firmly allied with the court party at this time.)
Suffolk probably negotiates for peace at any cost in the mid-40s, as he was instructed to do in OTL, and the loss of land -- some or all of which may have been granted to John in ATL -- may lead to Suffolk's downfall earlier in ATL. John would be the natural successor to Suffolk in ATL. He would pull York and other Normandy veterans into government. He would reclaim various land grants made by the court party, as he would understand how the crown's poverty had undermined the war effort in France.
Margaret likely never emerges as a major political figure in ATL. There is little evidence that she was an abnormally political queen before her husband's breakdown, and so I'd guess she has no quarrel with John taking control of her husband's government from Suffolk. The Beauforts are unable to maintain their hold on power without Margaret as a benefactor, and are removed from office. The WOTR are butterflied away entirely.
I’m afraid there is an issue with producing another son, with the same death as OTL is that Henry VI wasn’t born until 6 December 1421 at which point, Henry V was already back in France, following his return on 10 June 1421 and would never return to England.
We could maybe get around this without having them be twins, so long as Catherine is in Paris in spring 1422.
Henry V contracted dysentery at the siege of Meaux, 35ish miles east of Paris. The town surrendered on May 10. He died at Château de Vincennes, just outside the capital, on August 31. There could be some PR stunt where Catherine meets Henry as the conquering hero at some town between Meaux and Paris on his journey back to Paris, and they could conceive before his health collapses. But it would need to be a perfectly timed situation.
I would say in order John, Thomas, Edward, followed by George and Charles and maybe Richard (unless Henry was not a fan of the Lionheart)...
I
might flip Thomas and Edward in the order of likelihood, but otherwise I agree with you entirely.