Braeburn is probably the closest to Bramley that's easy to find in the US. Jazz is not a bad cooking apple either, but it's quite a bit sweeter than a Bramley. Neither one gets that distinctive Bramley texture when cooked though.
For a cooking apple, the texture is everything! McIntosh and Northern Spy are heritage breeds sold in the US that have good texture, though not so good as a Bramley. They can be hard to find, though, and usually expensive, whereas Bramleys in the UK are standard supermarket apples, available everywhere. I don't think I've ever seen a Bramley in the US; if I had room, I'd plant a tree.
