Hi, I was a member of Southbank Sinfonia in 2008, and several of my friends have been members, too. I'm from New Zealand and did the program before embarking on my Masters in the USA.
Firstly I would highly recommend the program. I can't think of a better way to improve your chops as an orchestral player. You get a lot of 'chair time'! They churn through a lot of repertoire and although it's hard work at times, I had a total blast. There is not a lot of time to practice outside of rehearsal, and it's a drag to lug your bass everywhere on the tube, but London is such a great place to be and the orchestra is really good.
In terms of visas, I did it on a working holiday visa. It probably wasn't the most efficient way to do it, but it was the easiest for me. I am not sure if that option is open to US citizens, you'd have to look into it. Regarding auditions, I'm pretty sure they accept tapes. There is an interview component that you can do over Skype I think. You do have to find your own accommodation, but when I did it, the organisation helped me to find a place to stay.
If you're after experience in a busy orchestra in one if the worlds great cities then I say go for it, it's awesome!