Hia Mick.

A couple of decent sized 'Balikbayan' boxes you can probably reckon on a £100-200 each. (Container, you're talking 4 figures).

It goes by size of box, not weight. The shipper normally supplies the boxes also.

I'm not familiar with Yorks area ones, as it's a long time since I've sent any, but perhaps start off an internet search with 'LBC' (one of the larger ones), just to get an idea of sizes and current prices.