We wound up using Anglo-Pacific. They provided the packing materials, collection (free on certain days of the week -- Tuesday and Thursday in Glasgow area), customs, and delivery. We paid for £550 for 9 boxes. They have a set sailing schedule which they post on their website so you have an idea of when items will be delivered. It will end up being a total of 6 weeks from collection to the US port (and I'm guessing another week or two for customs and delivery to our home in New Orleans).
The only thing that was confusing with them was their order form. It apparently is a bit confused and the form they read was different from the one I had submitted (though it could have been a complete confusion of people and quotations), so I had to give them the information a second time. Even then, neither of us realised the quotation was not for airfreight (as I originally submitted) until after collection. That wasn't a big deal, but it would have helped us in packing if we knew the time frame was 6-8 weeks rather than 1-2 weeks (some things were shipped which would have been carried with us if we knew differently).