The only version I know is on an old Topic Album "Transpennine" by Harry Boardman and Dave Hillery (12TS215). This track is Dave's. Sounds as though that might be the version you have. Certainly the first four verses are familiar though some of the wording seems slightly different and the conclusion is as you describe. Don't seem to be any intervening verses. The lad does not reveal himself as William but the lady recognises him from verse four as her long lost brother. I haven't played it for ages. Never heard it or come across it anywhere else, not even the tune. It's a Christmas Boxing song from Ripon. The sleeve notes refer to an additional "Lucky Birding" chant which might be the missing bit you ask about?

Lucky bu'd, lucky bu'd, chuck chuck chuck.
Master and Missis it's time to get up.
If you don't get up you'll have no luck.
Lucky bu'd, lucky bu'd, chuck chuck chuck.