A year 10 is going to add ZERO value to the BBC especially in one week.
I did mine at the BBC and was completely involved, writing scripts, interviews, booking guests in (famous ones) stuff that broadcast journalists do day in day out. I'm at university btw. If you do get BBC Work Experience, be pro active, they will not tell you or spoon feeed you like secondary school what to do.
With regards to food and travel, I hardly think a £20-30 travelcard and about the same maybe less for lunch is really going to break the bank. Some of you expect way to much, especially since it is them doing you the favour, not the other way around.