How do I find out who owns the rights to a very old song?

Specifically ‘Over the Hills and Far Away’ (of Sharpe fame) which was actually written in 1706.

I’ve spent the morning Googling for the existing rights but not found any definite leads. I’m guessing that the copyright on the song has expired?

Reason is I’m hoping to use it at my wedding in September, but they can’t play copyrighted songs…

I does not appear there is an issue. You are way past any conceivable copyright protection for that song.
How Long Does Copyright Protection Last?

http://www.copyright.gov/title17/92chap3.html

Very little from the 1800’s is still copyright. It would require a very long-lived author who wrote the material when very young. I’m going out on a limb and say NOTHING from the 1700’s is copyright. (Let’s say he wrote it at age 20, in 1899, and ded at 91, so that would be 1970. IIRC, plus 50 years, copyright until 2020. )

Some specific new arrangements of a song, or revised lyrics, might be copyright - but the original words or melody is not.

The composition itself is in the public domain, but any particular recording of a performance of the work is likely to be owned by someone.