If you're interested in the time it takes until the first byte of the actual media stream arrives, I think you'll need to do it in two steps, because the initial HTTP connection will only return a rtsp://... URI like this:

rtsp://rmlive.bbc.co.uk/bbc-rbs/rmlive/ev7/live24/radio2/live/r2_dsat_
+g2.ra?BBC-UID=b4361a170730f2451b5b6f95f05099551b1f970a5060d0735b4a906
+b12583889&SSO2-UID=
[download]

So you'll need to also connect to that URI in a second step (using the technique suggested by BrowserUk), and measure the total time of the two connects.  IIRC, the default port for the RTSP protocol is 554 (but be sure to verify that before trying...)

Update: actually, it seems you might need to jump through even more hoops to get at the actual stream...  Just read up a bit on that myself, and apparently there are other additional protocols involved: SDP (Session Description Protocol), RTP (Real-time Transport Protocol), RTCP (Real-time Transport Control Protocol), and possibly others... ;)

All in all, it might take quite a bit of reading until you're sufficiently knowledgeable to be able to low-level emulate everything that some media player program is doing behind the scenes... In other words, some more "end-to-end" approach might ultimately be more promising (like tracing network traffic with wireshark, or some such). But before going into that, I'd first spend some time googling to see if not anyone else has already figured it out and written a nice tool like ab (ApacheBench), only for benchmarking media streams... :)  Good luck!