It’s common practice to reduplicate words with slight variations to make up nonsense words; such as fuddy-duddy, willy-nilly, and of course dilly-dally. The root of dilly-dally is dally which means to waste dawdle or toy with someone's affections.
Lolly-gagging originally described as openly flirtatious love sick couples. Eventually, flirting gave rise to the what is today the most common sense of the word, to dawdle or dally . . . dilly-dally
So a dilly-dallying lolly-gagger could possibly be a time wasting flirt outrageously displaying his affections.
Or perhaps one who slowly sticks their lolly too deep into their mouth and gags.