Nope, it would have had the same meaning. ... of Lady McCartney's is what strict grammarians seem to prefer, and is probably to be recommended in formal writing like this. ... of Lady McCartney is the more 'natural' spoken version.
Thank you. Interesting answers - and an interesting question (for me, that is), because the double possessive, isn't seen in any other language that I'm aware of. Not that my awareness stretches very far.