Hi Rosaline,
Sorry to say that there's little hope of finding a birth cert. for your William since Irish Civil Records for Catholics start in 1864 and the Milltown Catholic Church records start in 1858.
1901 and 1911 Irish Census Records (the only Irish census records available) show Varilys in 1901 and Varrillys in 1911 (same family) in Carrownurlaur, Milltown, County Galway. Thomas the head of household was 70 in 1901 indicating that he likely was William's brother. The church records may still be the best bet if you have an LDS center near you to view the film. Since Thomas' oldest son was named William it is highly likely that Thomas and Williams father's name was William as the custom was to name the first son for the grandfather. As you may well know, a common variant of the name is Varley and there are Varleys in the Milltown today.

Hope this helps a little.

- John