If you get the iPlayer popup window, does it have a Play button? It should autoplay, at which point it turns into a Pause button, but I often find there's a noticeable delay before it starts. It might depend on the quality of your internet connection or how busy the site is.

If there's text saying "Low bandwidth version", it's actually a link to the low bandwidth version; the default is what they call Normal version, which probably requires broadband. My broadband sometimes goes up and down like a yoyo but without the predictability, and I find the low bandwidth version is less likely to suddenly stop and fail to reconnect.