OK, first up, Arlo's website has the 'official' chords on it.

http://www.arlo.net/resources/tablature/display.php?file=ring-around.chopro

But speaking as a rank novice, I found the D minor to B flat chord switch was a 
tad tricky which was a shame as this song's a real crowd pleaser!

Anyway, I found a way to cheat a bit and make the whole thing more novice friendly 
- transpose the chords down 5 semitones and capo it to the 5th fret to get it back 
into key.  It makes the chords way easier to play!

   C                 E7
1) I had a friend, a friend you could trust
Am            F           
Went into the park and got busted
C         D7                 G7
Doin' the Ring-Around-A-Rosy-Rag.
C                 E7
Went to the park, late at night
Am                        F            
Ring-Around-A-Rosy in the middle of the night
        C                 D7            G7   C    G7 
Yes, we all should do the Ring-Around-A-Rosy-Rag

Ch) Ring around, ring around Rose
Touch your nose and blow your toes
And mind after doin' the Ring-Around-A-Rosy-Rag.
Ring around, ring around Rose
Touch your nose and blow your toes
And mind after doin' the Ring-Around-A-Rosy-Rag.

2) We ought to send Officer Joe Strange
To some Australian mountain range
So we all could do the Ring-Aroud-A-Rosy-Rag
Would you like to put Philadelphia upthight, one mass
Ring-Aroud-A-Rosy in the middle of the night
Yes, we all should do the Ring-Aroud-A-Rosy-Rag

Ch)
C          E7       A7
Touch your nose and blow your toes
    D7                   G7
And mind after doin' the Ring-Around-A-Rosy-Rag.

Chords:
C     x-3-2-0-1-0
E7    0-2-0-1-0-0
Am    x-0-2-2-1-0
F     x-x-3-2-1-1
D7    x-x-0-2-1-2
G7    3-2-0-0-0-1
A7    x-0-2-0-2-0