DL being 5 hours faster would sell it to me, but I do understand the appeal of a 744+388 combination.
I think DL really improved a lot in recent years, to the point where I do prefer them over BA at the moment. As for crampedness, they certainly are better, albeit not by much.
I'm not sure how BA handles Y+, but I think they sell it as a separate fare class. With DL, you book a standard eco fare, and then get Y+ by paying a separate seat reservation fee for one of their "Comfort+" seats. Which usually is rather reasonable, and definitely worth the money on red-eye flights. Even if your company won't pay for it, you can still buy it yourself.
If you go BA, do get an exit row seat on the A380. BA has the same slave ship Y configuration as LH on the A380, which I find claustrophobic. On my only BA A380 flight (MIA-LHR), I had seat 80J (standard eco, uppder deck, exit row, aisle), and I think that is the best seat you can get in Y. Costs extra, though.
If you go DL, try to get row 29 on the 772. These are Comfort+ exit row seats. It's as close to business class as you can get without paying for it. If you don't want to pay for Comfort+, I can't really comment on the standard Y seats, as I've never been in one myself.
None are more hopelessly enslaved than those who falsely believe they are free.
(Johann Wolfgang von Goethe)