The simple answer is... ...it depends.
This is a issue about rights, not some abstract debate of what's "free" and what's not.
There are three types of rights. The question is about Distribution rights and who holds them for each programme. That will have to be known before any programme can be priced and sold for downloadable delivery.
1. CONTRIBUTOR rights - these are the royalties to the performers, paid through the various rights agencies. They are a continuing cost to the DISTRIBUTOR (as opposed to other production costs,which tend to be one-off costs during production).
2. DISTRIBUTION rights give the right to sell the finished programme across channels, territories etc, (according to what they agreed when they set the payment rate with their contributors - tricky part - for many older programmes, the idea of "online" or even "VHS video" wasn't foreseen at the time).
3. BROADCAST rights are permissions purchased by broadcasters from the distributors to show that programme X number of times on their channels. Nowadays "broadcast" rights need to cover lots of alternatives, online downloads, DVDs, or whatever. These are revenue to the distributor and on a good day are more than the costs.
Whoever wants to collect the £1.89s from the audience is acting like a broadcaster and will need to buy the online download rights off the distributor (who in turn, will need to shell out the contributors' cut).
The BBC often acts as some or all of these three rights holders, which is where the fun and games begin and helps explain the unique way in which the BBC is organised and managed - let alone funded.
What needs to happen is that ...
...for programmes where the BBC holds ALL the distribution rights (and we kind of guess they'd agree to sell broadcast rights to themselves, it would be perverse not to) then the programme is probably fully paid for and could be sold online, but with a 100% discount to those who show they've got a valid TV licence number.
... for programmes where the BBC doesn't hold and current broadcast rights, then it's up to the DISTRIBUTOR to licence the programmes for online download and the likelihood is the audience will have to pay the £1.89 because the BBC didn't buy all the broadcast rights in perpetuity that cover Online delivery.
... for programmes where BBC Worldwide has entered syndication agreements with others, then who knows??? All bets are off and many more lengthy meetings (with no biscuits) is the most probable outcome.
There's one tiny snag here.
It does rather depend upon being able to reach a definitive answer of who owns what across rights management in the BBC and that's not quite straightforward.
So the very old stuff, like Spike Milligan's "Q" series, is going to be hard. New productions will already have thought about internet delivery, but then it's just about how much the distributor should expect to receive for allowing someone else the right to charge the end-user £1.89.