Why do Oxford and Pembroke flee? Their army is probably intact and its support may be decisive for one of the other contenders.
They probably support a Regency for Warwick. I wouildn#'t give muvh fopr the boy's life expectancy, but he could serve as a stopgap while they all jockey for position.
Tudor's entire enterprise depended on him surviving. He goes down, the army routs (especially the French and Welsh contingents, hich means their army probably is not still intact) The Yorkists may have been divided but without their pretender in Tudor they will have no love for Lancastrians. There is too much support for other candidates for Jasper Tudor to get anywhere near the throne.
There is some confusion over where Lincoln was at the time of Bosworth but if he is in the North with the other Yorkist heirs, he is in by far the strongest place as President of the Council of the North to capitalise on the immobilised supporters of Richard in the area and to control Elizabeth and Warwick.
The main issue with this WI is that this all depends on how the battle goes down. If for instance Tudor and Richard are killed in the melee after Stanleys betrayal, it depends on what Surrey (now Duke of Norfolk) and Northumberland do. I personally believe Surrey would either flee to fight another day (against the stanleys) or if he gained Northumberlands support would take on the Stanleys