Trunks returned to his future before Cell Games?

[Hector]

After exiting the ROSAT (1st time) Trunks decides to just go back to his future instead of risking getting killed at Cell Games and never seeing his 'real' mother again, not even in the afterlife.

What happens next?

 

[Captain Cadaver]

Assuming this is immediately after exiting the Rosat rather than prior to his fight with Cell, this means Cell kills Vegeta and isn't given a reason to destroy the Earth immediately unless Kuririn tells him the same thing Trunks did. Assuming it's after the Cell Games are announced or that Kuririn fulfils Trunks' role there, the absence of Trunks and his death means Vegeta doesn't realise the importance of his family and probably doesn't adjust to life on Earth to the extent of his Boo Arc self, making him far less conflicted when becoming Majin Vegeta, meaning either he doesn't realise the full importance of his family until his Final Explosion or Goku decides to end things quick rather than attempting to reason with him and goes SS3, preventing Boo's revival. Assuming Goku doesn't and things go as they would normally between them, then the arc remains near identical from thereon since the plot would have Vegeta realise the importance of his family and respect for Goku regardless.

As for Trunks' timeline, him being far weaker would mean he'd be killed by Dabura even if he did access SS2. Whether or not Babidi ends up reviving Boo makes little difference as Zamasu would enact his Zero Mortal Plan in that timeline without resistance. Either he's eventually caught by the Daishinkan or Zen-Oh, or he succeeds in his goals. Assuming the former happens, the events of the Super manga remain unchanged as long as Goku and Vegeta still gain PSSB through training, otherwise Goku would be beaten by Jiren before he gets a chance to use UI. Due to Goku and Vegeta's gains from the anime version of the FT Arc not happening, either Jiren beats both of them and wins the ToP or general plot armour keeps things the same.