Maybe an earlier development of VHS so that it becomes commercially available in the late 1960's? The Beeb and other broadcasters see the commercial potential of shows like Dr Who, Dad's Army and Steptoe and they make a lot of money from video sales.
The problem with that is the expensive cost of the technology for the initial period, after which time it can then be sold as a potential consumer good, and the time it will take for the domestic consumers to properly buy playback devices like this earlier VCR. You also have to wonder at the culture of it, and wonder if they wouldn't just erase the master tape after it was on home video, or if they might just delete tapes without intent of putting them on video, whether part of the catalog of a show or all of it. It took some years into DVDs life before they actually sold entire series. Before that, it was just episodes and best ofs, and generally something like two episodes on a tape. That pick and choose culture will likely go into any earlier VHS just as it went into VHS when it did come, and such will be similar to the BBCs pick and choose of what to wipe and what not to wipe. You may get some more stuff saved because of that, but it still presents the problem that the BBC will just go through and erase tapes to save room.
The BBC is not alone in tape wiping/ tape reusing culture. America did it for many years. A lot of early American TV was lost, early Tonight Show episodes were lost, even the moon landing master tapes were erased (all we have are recordings from the view monitors). The BBC just seems to have continue the process well past when America stopped. You could have the BBC adopt that American perspective of preserving recordings earlier.