BBC social/political to-air radio pre-WW2 and during WW2 is said (Mass Observation? Angus Calders books) to have heavily influenced popular opinions on the war, and on current affairs, and arguably on things like religion/atheism.
"why we fight" was a constant of the BBC in these times. It probably influenced the end-of-war Labour victory over Churchill.
The BBC acted as a communications channel (one-way) to inform partisan networks. So, it directly influenced tactical outcomes towards strategic outcomes.
I would argue that mass communications effects on popular support is a form of influence to history. I suggest this applied to the Beveridge report, to postwar expectations of what housing should be, racism in Britain, attitudes to smoking, to the emergent understanding of the magnitude of risk in thalidomide.
BBC provided gaelic language services to the western isles of scotland, and so altered history by preserving non-english language in a small community.
The BBC broke ground by converting Hansard (post-event record of parliament) into live (real to-air recording of what was actually said) and so had a huge democratising influence.
More malignly, the BBC probably participated in the US/UK/AU repression in Kenya and Malaya, and the intervention to displace Sukharno in an anti-communist era. In the Vietnam era it acted more as social concence, but on some things, I suspect it fed the fever and helped hide the "black ops" side of what was going on.
I totally believe the BBC had a net positive influence on history, over time.