Could Nico & Percy become boyfriends - is this consistent with the story so far?

Yes I think it could and RR is an excellent writer who could carry this off.  I don't want to speculate about what the author might do in future stories but as I am old enough to be Nico's mum I wonder what I'd say if asked

Does Nico stand a chance with Percy?

I'd look at the facts.
1) In the real world, where boys are boys and not demigods with super powers, boys rarely stick with their first teenage girlfriend so Annabeth and Percy are likely to split up soon.

2) Percy doesn't seem to be interested in boys, in that way, yet.  BUT
I've known many gay & lesbian people who dated members of the opposite sex first, and were perhaps late teens or in their twenties when they feel in love with someone of their own sex.  My best friend at school had a couple of boyfriends and then, half way between her 16th & 17th birthday she met the girl of her dreams.  She said to me,
"being a lesbian had just never crossed my mind before."
As we were at school in the 1980s lesbians dating each other caused much outrage.

3)  Most boys are not gay so when if it comes down to statistics Nico should not pin all his hopes on this, even if Percy & Annabeth finish.

Would Percy getting with Nico be consistent with the back story?

Given points 1 and 2 above it is certainly possible but given point 3 above - my guess it this is an unlikely route for the story.

Dealing with the facts we know Percy is thinking of a long term future with Annabeth and has said they shouldn't be parted again.  Percy is growing up though, not yet 17, so young to make such life plans.

Re-reading the story you can see that Percy's great feelings towards Annabeth could be due to their close friendship over many years.  His love and affection for her could be as a friend rather than a boyfriend. Annabeth is in love with Percy and keeps saying so but that sentiment is not returned.

We often here the world "confused" applied to young people and their romantic feelings, a story in which Percy is confused fits well with all that has gone before.
The girls who may have been interested in Percy as boyfriend material have all been more aware of the situation than he has.

Personally, I hope this does happen or that the story is left open ended to suggest it might. What a great turn of events to this fantastic story if the central Hero turns out to be gay.