Goku says using the room too much can be damaging to the body and hurt training more than help at a point. If Vegeta went in for several days, it'd probably cause his gains and progress to stop after a while due to overtraining. Overall, I don't think anything would've changed for the Cell Games unless Vegeta and Trunks took it in turns, and even then they probably wouldn't achieve SS2 or reach FP Perfect Cell's level.
In the Boo Arc, with how there was no limit to save it for a desperate time, Goten and Trunks are brought into the Rosat by Piccolo and Kuririn from the start to perfect fusion. After one day, they'd be even stronger than in canon on top of having SS3 and all their techniques, so Boo is destroyed with little effort. Gohan still goes through Rou-Dai Kaioshin's ritual in the case of a new threat appearing and Goku remains dead.
For Super, Beerus wakes up and visits Earth. Without Boo, things go two ways:
1. Beerus leaves disappointed in not finding the Super Saiyan God. Freeza easily kills everyone a few months later.
2. Beerus gets bored and demands to learn about the Super Saiyan God. The group eventually consult Shenron and also wish to bring Tarble to Earth for the ritual, Vegeta being made the SSG. Things go the same up until Vegeta learns Freeza is on Earth. Without Shunkan Ido and needing to rely on Whis, everyone is killed before he can arrive, though Freeza doesn't destroy the planet due to wanting to torment Vegeta with this. Vegeta doesn't mess about in this version and kills Freeza once his stamina starts to wane. He then gets Whis' help to go to New Namek and revive everyone.
For the Universe 6 Arc, the lack of Goku means that Universe 6 wins the tournament due to Hit beating Vegeta and Monaka, though this doesn't really change much as the inhabitants of Earth shifted to Universe 6 would just use the Dragon Balls to make their new Earth vibrant.
The Future Trunks Arc doesn't occur as Trunks would've become strong enough from the Rosat training to prevent Shin from being killed and the lack of Goku means Zamasu isn't as stimulated to enact the Zero Mortal Plan. Goku not being there also dooms the main cast now in Universe 6 as Zen-Oh would just destroy Universes 6 and 7 at some point. The biggest change would be that, due to the main cast no longer being in Universe 7, Moro is unopposed until his actions become enough of a concern for Beerus to step in and destroy him.
