So, in this scenario, Edward III has only one son : Edward the Black Prince.
Since the Black Prince predeceased Edward III, his son Richard became heir to the throne then King as Richard II when Edward III died.
Things, however, will go differently from this point...
Richard II is likely to have a longer reign as he has no uncles and cousins who will rise against him (there will be Henry IV and then no War of the Roses), so he will probably rule longer. If he was impotent however, his succession becomes a problem has he will have no children.
The crown would thus pass to one of Edward III's brother or sister line. However, since Edward III's only brother, John of Eltham, died at age 20, he has no children. As for Joan of The Tower, the youngest of Edward III's sister, she had been to David II of Scotland but their marriage was childless. This leaves only Eleanor de Woodstock.
Eleanor had two sons who died prior to Edward III : Reginald III of Guelders (died 1371) and Edward of Guelders (died 1371). However, Reginald III had no children. I have no information on Edward of Guelders though, he might have children...
But, if this is not the case, then we have to go back one generation AGAIN, meaning to Edward I's children... I had discussed it on another thread considering the death of Edward II... But I have to update it since we are now considering this case.
Link to the thread for more details :
https://www.alternatehistory.com/Discussion/showthread.php?t=160895&highlight=Edward
One thing is sure : it end the Hundred Years' War earlier as the Kings of England will no longer have rights to the throne of France after Richard II's death (if the latter has no heirs).
The first line to consider would be that to Thomas of Brotherton, count of Norfolk. He had three children : Edward (who died at age 14), Margaret who died in 1399, making her still alive by the time of Edward III's death, and Alice of Norfolk (who died in 1352).
This would make
Margaret of Norfolk as the first in line for the throne. However, since she is a woman and is 57, I'm not sure she is in a good position.
Of Margaret's four children, only one had a descendance : Elizabeth (who died in 1368). Elizabeth de Seagrave had six children with John de Mowbray, 4th Baron Mowbray :
John de Mowbray, 1st Earl of Nothingham, who is 12 when Edward III dies but died in 1379, and his brother
Thomas de Mowbray, 1st Duke of Norfolk, who is 11 at the death of Edward III and died in 1399.
So, basically, the order of succession is the following one :
1. Margaret of Norfolk
2. John de Mowbray
3. Thomas de Mowbray and after him is bloodline