When Edward IV of England married a widow with two sons already, how did they figure out how those non-Royal sons would fit in the royal court?

Elizabeth's oldest son, Thomas, was about 9 at the time of her marriage to Edward. The younger son, Richard Grey (Elizabeth and Edward went on to have a baby together that they also named Richard), was about 7. Obviously no one ever thought these two boys should be in line for the throne, there was no question of their paternity. The advantages they would've been expected to gain by their mother's marriage were titles, wealth, and good marriages themselves.

Edward IV seems to have been a good stepfather, because they did receive those things under his reign. Their mother certainly advocated for them as well. Thomas, who would ultimately become Marquess of Dorset, made two good marriages, first to a wealthy heiress named Anne Holland who died young without issue, and then to Cecily Bonville who was THE wealthiest heiress in England at the time. Richard Grey was knighted, and served in several fairly important political roles under his stepfather's rule. It's safe to assume Edward IV would've gone on to do more for Richard Grey, and probably Thomas too, if he had not died unexpectedly in 1483 at the age of 40. He'd been a robustly healthy man and no one expected him to have such a short life, certainly not Edward himself.

After Edward IV died, technically his son (and the two Grey brothers' younger half brother) Edward V automatically ascended to the throne. But in practice, he was never crowned, and was illigitimized (along with all of his full-siblings) by his uncle, Richard of Gloucester*, who went on to proclaim himself Richard III. This was bad news for all of Elizabeth of Woodville's nine surviving children at the time of Edward IV's death - at least certainly for her four sons. The younger two, Edward V and Prince Richard, Duke of York, went on to become the fabled Princes in the Tower, disappearing from the Tower of England under Richard III's rule and never being seen again. The two older sons, sons of Sir John Grey, did not fair well either.

Sir Richard Grey was arrested just before his two younger half-brothers were taken to the Tower of England. He'd been escorting Edward V back to London in ceremony, everyone expecting Edward V to be coronated in the near future (Prince Richard was not there, but was soon also taken to the tower). He was executed by Richard III that year.

The eldest, Thomas Grey, Marquess of Dorset, was not involved in that particular exchange. But he joined the rebellion of the Duke of Buckingham against Richard III, and when that rebellion failed he fled to Brittany, joining Henry Tudor (who would later successfully defeat Richard III and then take the throne as Henry VII*), who had pledged to marry Elizabeth of York (daughter of Edward IV and Elizabeth Woodville, so his oldest half-sister; she was also the next potential heir of her father after her brothers were dead, but at the time no one had any notions of rising up in HER name against her uncle Richard III). Due to misinformation he received, however, he fell out of the good graces of Henry Tudor, and never recovered his former status under his reign as Henry VII.

To directly answer your question, this was the first case of a King of England marrying a widow at all, let alone one with children, so there was no precedent or tradition for that specifically. But the tradition for having step-children in general was to help elevate their status, perhaps less enthusiastically than you did your own children, but still you'd want to help your family gain whatever wealth or power or prestige they possibly could. So in that sense, Edward IV was a good stepfather to them.