Where did you get those words in the end? Are they compiled from fragments you found?

'The Carter's Health' is number 1384 in the Roud Folk Song Index. Four examples are listed there at present:

• 'The Bob-Tailed Mare'. Noted by Lucy Broadwood from a Mr Hoare at Houghton, Sussex, (October?) 1901. Tune and one verse printed in The Journal of the Folk-Song Society, I (4) 1902 219.
• 'The Carter's Health'. Noted by Lucy Broadwood from John Burberry at Lyne, Sussex, September 1892. Tune and text (verse and chorus) printed in Lucy Broadwood & J A Fuller-Maitland, English County Songs, 1893, 142. Transcription at Folkinfo: Carter's Health, with notes on 'hey', 'ree' and so on.
• 'The Bob-Tailed Mare'. Noted by George Gardiner and Charles Gamblin from Sarah Ann Goodyear at Axford, Hampshire, August 1907. Tune and text, unpublished.
• 'The Bob-Tailed Mare' ('The Carter's Health'). Noted by Clive Carey from Harvey Humphrey at Sullington, Sussex, 2 October 1912. Tune and text, unpublished.

Of the fragmentary references to be found via Google Books, most are to the Broadwood examples, though there are a couple I can't be sure of and others where the odd line is quoted in the course of narrative: the song -of which there was evidently no more than the lines we have here- was probably quite well-known in the southern counties of England a century ago. The entry in Dean-Smith, A Guide to English Folk Song Collections (1954) includes a cross-reference to a verse from a cante-fable in Bell's Nursery Rhymes With Chimes (1835). That verse is reproduced in The Journal of the Folk-Song Society, V (20) 1916, 327, but the words bear no resemblance to either of the Broadwood sets. (LEB herself thought them connected but didn't say why, while Miss Dean-Smith noted that the verse in the cante-fable had 'no obvious connection' with the 'Carter's Health' beyond sharing its tune; though there was a bob-tailed mare in the story). It's possible that the unpublished Gardiner set may be connected to the cante-fable verse, I suppose; it begins differently from the others. I don't think it's one I've seen in the MS collection, and there's a chance that it isn't really related to the other three at all except through the title.

The three 'versions' posted in the Forum are virtually identical to those here. Actually they are all the same one, posted three times by the same person, who learned it from folk-club friends; I'm pretty certain that it's not an independent variant but one of the two Broadwood sets, with original 'hoo' rationalized to 'whoa'; but without the music I couldn't guess which. I'd also expect that the 'Tale of Ale' version is one of the Broadwood ones: again, the tune would decide the matter.