Different tickets but same PNR is not really that odd at all.

If DL says "yes" that is the most important if you are starting your journey with Delta.

Delta should do it (and BA should do it on the return trip too). But I always get to the airport early then be polite but persistent. The airlines have an interline and baggage agreement with each other.

And being on the same PNR has the advantage that DL should see the BA flight in their system, thus it should be easier for them to check the luggage to LCY (if separate PNR's then DL would need to type in the connecting flight info manually, but that shouldn't be necessary in your case).