No, it was almost always December or January. Some exceptions were made to award it in autumn, but don't remember any September ones. Something like late November, and maybe during the merger they had given some awards in October or November as well.
They usually announce the lists in September or October, so it's unlikely that they gave the awards then.
Either way, you're not wrong on that, but the thing is, you can be out of CL or on your way out of the groups and still win it in the season that the award is given. But as you said, it's the previous season that matters.
And Lukaku is nowhere near Ballon D'Or level, so I'm not even sure why people are even considering him for this. Belgium going far will probably just give De Bruyne better odds. Lukaku may end up getting a vote here and there, but last season he had more chance imo to be involved with a EL runner up campaign where he scored in every round, even if he got knocked out of the group a few months later.
CL, international tournaments (typically European related ones) and hype is what makes you relevant here. Lukaku may have some hype due to English media traction, but he's nowhere near that level. But if someone like Kane gets consideration, I don't see why not, but it also doesn't say much. Getting on the list due to hype is not something to be really proud of.
In 2019 we had the likes of Joao Felix, Sterling, Aubameyang, Koulibaly and Benzema as part of the list that didn't really do well in CL.