In the end, I flew SQ and AA, where SQ had a wall mount bassinet (but the seat is large enough as a bed to accommodate both mother and baby in a travel bassinet (Eddie Bauer one). On AA there was no bassinet so we used the travel bassinet on the floor, which wasn't really used much as he slept with one of us during most of the trip.

I did more research on the UA bassinet and essentially, found that it restricts the business class seat recline when it's on the floor. We did this trip when our baby was 2 months old and quite frankly he was breastfeeding so often for most of the trip it wouldn't have mattered if we had a bassinet or not.

Fast forward 4 months. We flew again when he was 6 months old and again on SQ and AA. SQ was great as usual with the large seat, and since this is always an overnight flight for us, it's great. The baby didn't want to sleep on our flight to DFW from NRT so the bassinet was not used. We did try the bassinet on the SQ flight and he really didn't like it, as he is normally in a crib. On the return flight at 7 months, he slept very well on the floor with the travel bassinet.

Some airlines won't allow the bassinet on the floor, as we found on a recent flight with SQ, KIX-SIN in economy, but SIN-NRT was okay and it worked great. I wish I had more info for you on the UA seat and bassinet arrangement.