Tricky one, is taking a refund and rebooking elsewhere feasible? This will be by far the easiest option.
Otherwise, BA are unlikely to voluntarily rebook on any of the direct options for a schedule change. They will likely claim you can only choose between a refund, rebooking to alternative dates where they're still running, and alternative rebooking options that fall within their
standard guidelines (i.e. in this scenario that'd be Oneworld - QR via DOH? - or possibly LH group, though I think the latter agreement may be only for disruption on the day?).
However this is not compliant with their UK261 obligation to rebook you either at the earliest opportunity or on another date of your choosing, "subject to comparable transport conditions". CAA guidelines suggest this implies a requirement to offer rebooking on other airlines if they cannot offer a comparable same-day alternative, and that seems to be giving a considerable degree of leeway to the airlines in my view.
I would engage with BA in good faith, giving them an opportunity to rebook you on any alternative direct flight the same day (e.g. FB). When/if you get the inevitable rejection/refusal, leave the booking as it is (i.e. don't accept the infeasible rebooking suggestion or take a refund) and send a Letter Before Claim to BA's legal address informing them of your intention to rebook on OAL and recover the cost from them, if they continue to refuse to honour your UK261 rebooking rights. That will put you in good stead if you do need to bring a CEDR/MCOL claim against them - you will have given them every opportunity to honour their obligations and avoid being liable for your expenses.
Then if it does come to it, rebook on OAL, ideally with screenshots of the likes of Google Flights to show that the price/times are reasonable based on what's available. Keep the receipts and submit your expenses claim to BA. BA may well refuse to process it until after the booked travel date (and TBH it's now too close to the travel date to be worth spending weeks/months arguing the toss over that). Keep screenshots of the confirmation page when you submit your claim, and the summary of what you sent.
You can then proceed to CEDR 8 weeks after submitting your claim, or once you have a "final response" email from BA. Alternatively or if CEDR rules against you, it's going to have to be resolved through MCOL (i.e. Court).
As you can see, forcing through your rebooking rights by doing so manually and claiming it back is the far trickier option. You also need to make sure any further sectors in your SOF booking aren't cancelled off for being a "no show" (though a non-accepted rebooking should remain on "HK" status in BA's system, meaning this shouldn't happen).