I don't know what Pete Waddon meant; maybe he wrote a new tune for it. Or maybe he wrote additional words, not given here. The words we have quoted are at least 140 years old:

From London Cries: With Six Charming Children by Andrew White Tuer (London: Field & Tuer, 1883), page 13:

One of London's best-known characters, the Waterman, does not appear to have adopted a cry; or, if he did, no mention of it can be found. But a correspondent of Notes and Queries (5th S. I. May 2, 1874) says: "I heard this verse of a very old (waterman's) song from a very old gentleman on the occasion of the last overflow of the Thames:—

" 'Twopence to London Bridge, threepence to the Strand,
Fourpence, Sir, to Whitehall Stairs, or else you'll go by land.' "

The point of departure, however, is not given.