I would imagine most of these navvies would be English speaking immigrants in which case they would be mainly singing the popsongs of the period. In 1850 the biggest craze sweeping N America and Britain was the minstrel troupes like Christie's and the Virgina Serenaders.
If you just require texts the Bodleian Broadside Ballad Website will give you tens of thousands to go at. If with scores the best bet is The Lester Levy website. If you're looking for home-made and folk-songs no-one was really recording this in 1850 so the broadsides would be the best indication.