Yes, in the circumstances you describe there would be two separate claims - one for cancellation, then one for delay. There is a CJEU judgment (Finnair C-832/18) which explicitly confirms that you are entitled to make multiple claims in relation to the same journey - effectively one for each set of flights you are booked on, and which are subsequently delayed or cancelled.

BA probably doesn't receive many claims under such circumstances and I don't think their processes are designed to handle such cases (the same goes if you are disrupted on both the outward and return legs of the same booking). If you submit multiple claims then the second claim will likely be silently closed as a duplicate behind the scenes, due to having the same booking reference number.

So it'd be highly advisable to take a screenshot (or print to PDF) the page just before and after you submit the claim forms, so you have clear evidence of the contents and date you sent in your two separate claims. Do try to keep the freetext section of each form brief - 1 or 2 sentences max (e.g. "BAxxx cancelled, rebooked onto BAxxx arriving X hours later") - as they can look up all of the necessary details from your booking reference. Briefly worded claims seem to be handled more quickly.

If BA refuses to pay two lots of compensation (without a valid reason) and confirms this is their "final response", or once 8 weeks have passed, you can proceed to CEDR, the ADR scheme that BA are signed up to. I would set yourself a reminder for this, or if you have Gmail I find it's convenient to use the "Snooze" functionality with an appropriate date set. CEDR isn't quick either, so you want to get the ball rolling ASAP.