Well if Eastenders was to do a cross-over show with Spooks and enough people wanted to watch it online on BBC One at the same time it might happen.
Live streaming always carries the risk of jitter as connection speeds go up and down, and while you will need something like a 1Mbps or faster connection to watch the live feeds, the bigger issue is whether your provider can cope with the hundreds, or thousands of others watching online at the same exact time. Shows like Eastenders and Strictly Come Dancing have an audience of millions, so only a small proportion need to watch it live to chew up a lot of providers bandwidth.
While the main focus for many is on getting the speed of the last mile of their broadband connection running faster, be it copper twisted pair or co-ax, there are those already finding out that this is not always the bottleneck. Once your data hits the first cable cabinet or telephone exchange it is on a shared medium and if more than 2 or 3% of people are busy actively using the connection then things will start to slow down. Some will try to say this is unique situation to the UK, but it is not, broadband worldwide is cheap simply because of the shared nature of the capacity.