In my searching, I stumbled across a Walter Baskval born about 1806 in Ireland. He turned up in the 1841 Scotland census, living in Lanarkshire. He was a private in the 17th Lancers. There was a Lucinda Baskval born about 1806 listed in the same census, as well as three children with the last name of Baskoal: James, John, and William. All born in Ireland.
I don’t know that this is at all helpful in any concrete way. I have no proof that these two names are variants of Basquill (though how could they not be?), nor can I connect the names and dates to any Basquills I know about. It’s conceivable that this Walter–and that’s not a very common name in Ireland at this point in time–is Walter’s grandfather. If so, he’d be my, let’s see, great-great-great-great grandfather. He could also be my Walter’s uncle. Or he could be no relation at all, though I find that unlikely.
The real question, though, is where the heck did this variant of the name come from? I wish there were a scan of the census page available, because I’d like to see the actual handwriting. Sometimes, weird variants are due to bad transcription. Other times, they might be due to the enumerator not having a good ear. This is especially problematic if the enumeratee is illiterate or not fluent in the enumerator’s language, which is a distinct possibility in this case. Walter could well have been unable to read or write, or he may not have been very fluent in English, if Gaelic was his first language.
As for the different spelling between the parents and the children, I wonder if they were enumerated separately? The kids were pretty young (1, 4, and 5 years old), so they were certainly unable to read and write. They also may have been in the care of a neighbor or someone who didn’t know how the name was spelled. Without seeing the actual census form, it’s impossible to guess.
I also found a Peter Basguill born 1801 in County Mayo, Ireland. Peter was in the 1851 Scotland census. He was a grocer in Fort-William, Inverness. Possibly a brother of this older Walter? Possibly the father of Michael? Or, even, the grandfather of Michael? As far as I can tell, this is the only instance of this variant of the name showing up in the UK census records.
And just for fun, I tried Google. I got no hits for Baskval and Baskoal and only three for Basguill. Two were for a Martin Basguill who is buried in Pennsylvania, and the third was for a patent application.
red paper boat 
You are rightly confused.
Lucinda Baskval was the first wife of Walter Baskiville who moved to New Zealand in 1851. William, John and James are their three children.
Lucinda died and Walter re-married.
His name was extremely fluid – when he joined the 17th Lancers in 1827 he called himself Basval, but over the years Baskivil, Baskivile, Basquill, and Baskeville are used in various documents.
He married a second time in 1848 to Anne or Anna Maria McFall of McFaul.
The New Zealand family all used Baskiville but census results and electoral rolls have Walter using several variants. He has Baskiville on his grave.
Walter’s grandson Albert Baskiville did publish a book on rugby as Baskerville just to add to the confusion.
Of the three children mentioned above William and John seem to have come to New Zealand. I fear for young James… William tragically died of TB soon after arrival in Auckland. One history has John joining the army in New Zealand.
Sorry if this adds to the confusion.
LikeLike