I've sent a MIDI of all four parts to Joe for posting here. The reason for that is so that people who don't know the song can hear what it's like. It also makes sure we are all talking about the same melody. (I did that before I saw Q's link. Thanks, Q.)

It's a fine song. Thanks for bringing it up. Here are the chords. There are lot of them, but they seem to be needed. For one thing, they are there in the parts. I've marked some that you might not really need with a *.   Good luck with your ukelele!

D Be-hold that
AD star!
G Behold that
Em star
G up
D yon
Em -der.
D Be-hold that
AD star!
D It is the
G star of
D* Beth
A-le
D-hem. There was no
A room
G found
D in the inn. This is the
G star of
D* Beth-
A le-
D hem. for Him who
A was
G born
D free from sin. This is the
G star of
D* Beth-
A le-
D hem.
D Be-hold that
AD star!
G Behold that
Em star
G up
D yon
Em -der.
D Be-hold that
AD star!
D It is the
G star of
D* Beth
A-le
D-hem.