Please look at all the notes on the C space of the staff.  They are all accidental sharps.  Someone could put a sharp at C in the key signature and remove all those accidental sharp symbols, making the key signature that of D major.  Keep in mind that whoever wrote the score may lack understanding of such things, or may have been careless about it.

Do the notes of the chords stay in the D scale?  If not, do they seem to agree with the accidentals in the measures where they are used?

I think you may get confused if you expect a usual chord progression in this piece, relative to D.  Most notably, it doesn't end on D.  There are ways to adjust your thinking to that, ha ha.  To give you a hint, I think it's going to be best to think of the piece as being neither major nor minor.

BTW, I played the melody as written, on violin, and I found four notes that didn't seem to agree with my memory of the tune.

I figured some information about the chords, using the chord finder at http://chordfind.com/ .  I'm not sure of the implications, so, if you have some analysis, please contribute it.

The piece has six chords: E7, A7, D7, G, B7, Em

I think we are going to agree that the tonic is G.

I compared some of the facts seemingly related to the choice between key signatures G and D.

Note that the piece does not contain a D-major chord.

For each chord, I asked whether it fit perfectly in the candidate key signatures, that is, whether all the notes of the chord are in the key signatures.

E7,G,and Em fit both G and D.

D7, having a C, which is out of the D scale, fits G alone.

A7, having a C#, fits D alone.

B7 would fit both except that it contains a D#, which fits neither.  There is one D# in the melody.

D7 is the dominant seventh chord (V7) if the key signature is G, and D7 is followed immediately by G (the tonic) in all three places that D7 occurs.  G is preceded only once by anything other than D7, namely B7.

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I'm thinking that the repeated V7/tonic sequence and the absence of the D-major chord are the strongest indications that the scale should be considered to be G-major rather than any mode of D, this despite that two sharps reduces the number of accidentals needed.

After this analysis, I am less inclined to say that the score is a mess.  But I do think that some notes in the melody don't follow the way the tune is usually played.

Edit: 12:50 pm ET Sep 20:  Notice also that the piece doesn't have much of any non-major sound to it.

@Fiddlerman Thank you, FM.  It's very gratifying to get such a compliment from you.

@DanielB Same kind of thanks to you, and the same feeling.

I just learned the importance of the dominant seventh cord a couple days ago, watching a youtube lecture video on the sonata form, where it was stated that V7 is always or usually invoked near the end of the development section of a sonata.  And luckily I was chatting in the company of a new member who informed me that V7 cries out for a return to the tonic, implying that the recapitulation and its return to the home key will be all the more satisfying to the listener.  Now would anyone be surprised to know that that member is also our Sweet Georgia Brown expert, having played it on about five instruments---sometimes, I believe, in paid performances in some pretty large venues?

I would be very happy to get some more analysis of this piece, agreeing or disagreeing, especially about the notes of the chords that don't quite "fit".

Steve said
Stonedog's version is in the relative minor key of E minor.

To be brief, that's probably not the best way to think of this piece.  Notice that both the melody and the harmony end up on G.  Also notice that the tune doesn't have much of a sad sound.

G major = E minor.

True in the sense that those two scales each can easily be written using the same key signature: one sharp.  Also true in that a tune in E-minor and a tune in G-major are both best written with a key sig of one sharp.

I love to see someone starting to understand key signatures and scales.