Here is my ABC essentially following Karen and other internet sources:

X:1
T:Alfred the Alligator
P:Round for (up to) 16 parts
C:(Author yet unknown)
M:4/4
L:1/16
Q:1/4=160
K:G
"1." d2 d2| d2 d2 B4 c4 "2." A4| B4 G4 A4 "3." D4| \
w: Have you e-ver been down the wa-ter-spout? Right
G4 G2 G2 A2 A2 "4." c2 c2| B4 G4 A4 "5." D4| \
w: down to the bot-tom of the wa-ter sys-tem?
E4 E4 F2 F2 "6." F2 F2| G2 G2 G2 G2 "7." A4 A4| \
w: There you'll find a litt-le al-li-ga-tor that goes
B4 B4 c4 "8." c4| B2 B2 G2 G2 D2 D2 "9." F4| \
w: by the name of Al-fred. If you see him, he's
G4 z4 z4 "10." d4| B4 G4 z4 "11." d4| \
w: mine. I lost him. I
d2 d2 B2 B2 c4 "12." A4| B4 G4 A4 "13." D4| \
w: kind-a threw him down the wa-ter-spout, and
G4 G4 A4 "14." c4| B4 G4 D4 "15." F4| \
w: now I'm real-ly lone-ly, 'cause he's
G8 z4 "16." d4| B4 G4 z4:|]
w: gone. I miss him.

I think it was in the 1980s when I read that in NYC young alligators were the latest ladies' fashion pets. When they grew too big, many owners flushed them down the toilet. In the sewerage they proliferated, feeding on the rats. It sounds like the prototype of an urban legend, but was reported for a fact.

Leenia, of course you're right, but most children will detect the ironic self-criticism in the phrase "I kinda threw ...". With a little help, they will learn by the bad example that owning a pet is a responsibility.

In my opinion, the composition (as a round) is certainly not the pinnacle of sophistication, but the idea and the lyrics are a notable achievement, fun for children and adults.