I haven't used EyeTV, so some general knowledge on TV, and some guessing.
Obviously, you're not going to get any US broadcasts. As the UK, it should work fine with whatever is coming out the pipe.
As for the internet based stuff, you're going need the US IP to get access. Maybe someone has done a script to turn on a VPN when trying to access those sites for EyeTV, but I suspect not likely.
I do know of a service that is online based that will give you access to US TV broadcasts. Check out USTVnow
. It's specifically for US people living abroad that still want TV from home. The broadcasts are based out of Philadelphia. You don't need to use a VPN. It has a free month trial. The free service is only ABC, NBC, FOX, CBS, and PBS with low-res only. They have a plan for $30 a month that gives hi-res, and more channels like Disney, ESPN, Discovery and so forth. They even have some kind of DVR function.
If you really want to have it all, there is another company they work with that you can get something like two hundred channels. I think they have some kind of plan for premium stuff like HBO. I don't remember the costs, but I think one was $50/month, and another being $99/mn. I believe this company offers HD.
If you have a good VPN service, I would see if you can use Hulu or Netflix or other such services, and see if they have the shows you want. Most of those sites offer HD, it's just not the clarify of OTA or cable/satellite. It costs some money for a good VPN, but probably cheaper than using that other online TV service.