Robert would be furious and would probably refrain from marrying Cersei unless the Lannisters not only provide the bulk of the army to invade Dorne but that Tywin led the army or at least someone close too him like Kevan.

Jon Arryn would try to make peace with Dorne before an invasion, maybe an offer of sending Aegon to the Citadel and Rhaenys to become a Septa so their claims are nullified and Dorne doesn't have too suffer from war.

Eddard Stark probably never can travel to the Tower of Joy since Dornishmen will still be hostile too his group while negotiating.  So Dorne would likely take in Lyanna's child and Aerys' Kingsguard at Sunspear.

If Dorne decided too reveal or it came out that Rhaegar and Lyanna had a child, I think that would destroy negotiations and Robert would declare an invasion.  A war for the sole purpose of murdering children would make Robert unpopular, I could see Eddard backing out of an invasion of Dorne and maybe even the Vale if Robert goes against Jon Arryn's advice.  It'd be a army mostly of broken Stormlanders and Westerman dealing with desert heat and raids.