Without a doubt they were influenced by Lonnie - they were prompted to form a skiffle band because of him!
I was never much of a fan of this type of music but my Dad is! He was a huge fan of The Chris Barber Jazz Band (of which Lonnie was a member) and later Lonnie Donegan and Ottilie Patterson. He totally dismisses Lonnie's novelty records though - such as "Does Your Chewing Gum Lose It's Flavour" and "My Old Man's A Dustman".
here is a recording, taped on the day John first met Paul (July 6th 1957), of The Quarrymen performing a Lonnie Donnegan track called "Puttin On The Style". It is very poor quality but John's voice is still very distinctive.
"Puttin' On The Style" July 6th 1957 [music]http://h1.ripway.com/michellerevolution/PuttinOnTheStyle.mp3[/music]