very sorry for my english,but I decide to answer on this and explain how I see the situation:
There’s no reliable information in canon about Sauron’s fate after he lost Tol Sirion. Sauron was defeated and wounded, he left an island and ran to Taur-nu-Fuin - and that’s actually all we know.
So after all this events I see at least 3 (actually 5,but I will describe only 3) possible versions:
1) Sauron could stay on Taur-nu-Fuin, because he feared of punishment and became an Ungoliant-type of evil. And he didn’t show himself during War because he was on his own side.
2) Morgoth could call Sauron to account and he would return to Angband,begging for mercy. After this Morgoth could banish him - in that case Sauron would become an outcast and probably wouldn’t take part in war.
3) Script#2 with one ”but”. Sauron was Morgoth’s most powerful servant and it was not good to stay without him during infinite war. But loss of silmaril had a great impact on Morgoth’s position of invinsible king and Sauron was indirectly guilty in this situation ( at least from Morgoth’s point of view). So until the WoW Sauron stayed with Morgoth, but in ”disgraced” status. This version is one I use.
Actually I can explain my point of view better - but not on english, unfortunately.