"Sumer is Icumen in" is probably the oldest English song you are likely to actually hear anyone sing.

I've never thought of I live not where I love as sounding particularly old. I'd have guessed 19th or 18th century, though I see from a glance at earlier threads that it's got some 17th century roots.
....................

Looking for a favourite Tudor is a bit like looking for a favourite Don in some Mafia family. I'm sure that if Henry VIII had actually written a great song, he'd have made quite sure we all knew about it.