Chant is old as an English word, used by Chaucer (1386, Miller's Tale, 'chanteth').

Nordhoff (1856) was the first to use the ch-form for a sailor's song, as "chantey-man" for the leader of the work songs.

'Shanty' applied to a sailor's song first appeared in Chamber's Journal (Dec. 1869). "Said to be a corruption of the French 'chantez' ...."
(Above notes from Oxford English Dictionary)

This has all been gone over in previous threads.

Shanty should be reserved for the original Canadian and U. S. meanings, "an establishment ... organized in the forests in winter for the felling of trees; later something built of lumber, usu. temporary. Corruption of French 'chantier'. Canadian Dictionaries.
In U. S., first mentioned in print in 1820, a hovel called a shanty, "somewhat in the form of a cowhouse."
Oxford English Dictionary.