I'll try and answer your questions Tarian.
BBC local radio audiences are measured by reach (ie proportion of total local population who listen at some point each week). This varies massively, depending on geography. 3% in London, 14% in Kent, 20% in Nottingham (my nearest one), 35% in Ulster. Typically 12-20% across the country. That's many millions of people. Mainly it's for the news-based breakfast shows and local sport, weather and travel. Local radio really comes into its own during local emergencies - snow, floods, school closures etc. Instant info that no one else is in a position to provide. This is all public service stuff. BBC local radio is actually part of emergency planning systems for that reason. So if it's lost, there'd be a big hole no one else would fill.
9am-4pm (ie until the news-based drivetime shows) is more chat and music which you could argue is less distinctive. But no radio station can build an audience unless it's broadcasting across the day. In my view, after drivetime you could save money by having a national shared show or even switching off the output completely (unless there's a midweek game, perhaps). For those who listen, it would be a massive blow though.
I take your point about radio being funded by the licence fee. You have to remember this was set up before commercial radio existed, so it is an anachronism now. But what's the alternative? Advertising? Local commercial stations can't generate enough income, which is why they've been absorbed into national networks. Someone has to do the local stuff. I'd guess that all 39 BBC local radio stations combined cost less than ÂŁ200m a year (when I ran various ones 2001-2005, my annual budget was about ÂŁ2.5m - for staff, buildings, engineering etc). They're a tiny drop in the BBC ocean cost-wise.
One final thing. Did you know that local newspaper democracy reporters tasked with covering councils, NHS etc are paid for by the BBC? Those roles would be lost too if the licence fee goes.
PS My favourite station for away games is Radio Sheffield and their "praise or grumble" post-match phone-in. Listening to local fans having a grumble after we've tonked their team is hugely entertaining!