All these "Cuddy"s sent me looking for answers. I found this:

CUDDY
Possibly derived from the Persian word ''gudda'' or the Hindustani ''ghudda'', a cuddy is a horse - not a thoroughbred racehorse, though, but a more workaday quadruped. A joiner's sawhorse was often referred to as a cuddy, and a gymnasium horse was also a cuddy to schoolchildren. To give a child a ''cuddy-back'' is to carry the child on your back. An old childhood rhyme used to go - ''Matthew, Mark, Luke and John, haud the cuddy till I get on!''

Though someone claimed in the "Kitty ALone" thread that a cuddy is a frog, I can't find any confirmation of that. The above is Scots, and presumably also Geordie. Other sites say that a cuddy can be a donkey, too.

Cuddy is also the diminutive for Cuthbert.

So this song is a play on words, right? Bess is looking for Cuddy (Cuthbert), the singer thinks she's looking for her cuddy (horse or donkey).

~ Becky in Tucson
(wading through the Geordie is rough going!)