'Toot sweet' is a English slang term meaning 'quickly' or 'at once'. It is a corruption of the French 'tout de suite', which literally means 'at once'. This use was popularized during World War I, when British and American troops in France began actively mauling the French language. This is an example of Tommy's French, French as spoken by the the average soldier. Other examples include napoo (il n’y en a plus), comprey (compris), and boche (no one really knows).
Toot Sweets, with the plural 's', is also a song from the musical Chitty Chitty Bang Bang.