Boys:
Henry Connor - Nice combo, great flow. I think Connor gives Henry a bit more spunk and slightly modern feel.
Henry James - Short simple and to the point, I feel that James as a middle name is a bit filler but it still sounds wonderful and if you love it then I wouldn't let it affect my decision. Has a very classical feel.
Henry Elliot - <3 My favorite of your Henry options. To me this name sounds smart and literary. LOVE IT!
Charles Ellery - I adore the nn Charlie but dislike Charles because of several people I know with this name. I like Ellery a lot (could have something to do with Eller being my favorite boys name

). Overall I think Charles Ellery is a great name, classical with an eclectic twist.
James ? - Hm. James used to be one of my favorite names but I'm liking it more in the middle name spot these days...Hm, what about...James Whitman, James Caspian, James Alistair, James Alfred, James Dickon, James Oliver, James Bennett, James Maxwell?
Girls:
Lily Paige - Okay but a tad blah.
Lily Katherine - Lovely. I think Katherine lends the sweet and cutsy Lily a bit more maturity.
Lily Charlotte - Gorgeous. Lily Charlotte is classic and beautiful and has a softer feel than Katherine.
Lily Nina - Not sure this one flows very well. Would you say this NEE nah or NIE nah?
Paige Marie - Not a fan of Marie, very much overused in the middle name spot in my generation.
Natalie Jane - I like Jane, simple, sweet and underused.
Natalie Brooke - Brooke is a bit overused and I've never been a fan. I much prefer Jane.