Will the BBC version of the Life work correctly on an American Blu-Ray player?

I don’t think it would play as you are in a different region. You can contact the BBC Shop to find out;

Email: bbcshop@bbc.co.uk
Telephone: 8.30am-6pm Monday to Friday
Outside the UK +44 (0)1795 414 989
Fax: +44 (0)1795 414 555
Postal address: BBC Shop, PO Box 308, Sittingbourne, Kent ME9 8LW

Or BBC America

I am not the expert, however if you output with HDMI from player to TV HDMI. There should be no “region” issues.
HD to HD should work fine.