Much depends on how competent Thomas is, and what his relationship with his brothers is like. Regardless, whatever his faults, Thomas will likely be a far, far more competent king than Henry VI and probably more active/successful in stamping down on any discontent (or at least not letting royal authority break down as completely as it did under Henry VI).
As for other Plantagenet claimants- Edmund Mortimer, 5th Earl of March, was loyal to Henry IOTL after being used as a pawn of his uncle and the Percies in his youth- he obviously informed on the Southampton plotters. So it depends on if his loyalty transfers to Thomas and the Lancastrians more generally. He doesn't seem to be overly ambitious.
That said, Thomas was supposedly hotheaded (precipitating his death at the Battle of Bauge) so I suppose it's possible he could alienate Mortimer and trigger a rebellion. Say, people plot on Mortimer's behalf, Thomas denounces Mortimer, Mortimer flees, rebellion. Though Mortimer's health is far from robust- he got dysentery at the siege of Harfleur and ultimately died young, so he mightn't necessarily last too long.
Elsewise, does the Duke of York still die at Agincourt? Because if so, his nephew and heir Richard is 4 and unlikely to cause trouble (and King Thomas might be able to deny Richard his inheritance because of his father's treason).
Also, I think Thomas is already married- to Margaret Holland, and has a bunch of Beaufort step-kids by her. But Margaret's father and brothers are already dead, so this marriage doesn't provide him a great deal of domestic support. IIRC correctly he was relatively close to his stepsons IOTL- John and Thomas Beaufort were captured after accompanying him at Bauge- so when they're grown up they could bolster his regime. He's also got two step-daughters to perform diplomatic manoeuvres with.
Of course he didn't have any kids by Margaret IOTL, despite a decade of marriage, so that could cause headaches down the line.
So, there might be some quarrelling with his brothers (John and Humphrey obviously didn't get along IOTL), but without a 15-year minority followed by a terrible king the domestic situation probably doesn't go as thoroughly to shit as it did IOTL. But with defeat at Agincourt (and the death of the English king) and the need for Thomas to sort things out in England (at least in the short term), then the English position in France could crumble.