So far I have discovered the following:
1) Laezel - You don't have to do anything special, just raise her approval and she'll offer you sex. Right in act one. But if you have already started a relationship with Karlach, she will reject you with the reason that she is only interested in "fresh meat".
2) Shadoheart - You must have a high approval and you must not attack the druids in act one. Then you can start a romance (literally right after celebrating the destruction of the goblin camp.)
3) Atararion - You don't have to do anything special, just raise his approval and he'll offer you relationship. Right in act one.
4) Gale - You don't have to do anything special, just raise his approval and you can start relationship. Right in act one.
5) Karlach - You don't have to do anything special, just raise her approval, complete both quests and you can start relationship. But if you're in a relationship with Laezel, you won't get the option to start a relationship. Dump Laezel to get that option.
6) Wyll - No idea. Not even a hint of romance in the first act, despite the maxed approval.
If anyone has other experiences, I will be glad if they share.
How early do you have to dump Laezel? I tried reloading to near the end of Act 1 and doing it and nothing changed.
I’m interested because I’m not sure it will make narrative sense for Karlach to be with a Chaotic Neutral Dark Urge monk in my second play through. More sense for Lawful Good Sorcerer Tav.
EDIT: You have to do it before a certain pint in Act 1 and rest. Confirmed working. Weird mutual exclusivity though. It would make more sense for Shart and Lae to refuse each other because Shart says so to Wyll.