We flew overnight March 1st>2nd and it was fine, generally--
I had prepped Mrs.Rat for a degree of confusion and oddness (I had lived in Tokyo in the 1990s, before Google translate or Citymapper and when there was much less English signage) and basically told her that anyone with a hardhat, hi-viz vest or a little light-up stick must be obeyed and any time spent trying to figure out why we were being told what we were being told was not going to be a high-quality use of time or brain-space. Just do what's told. With this in mind, we got through immig without much fuss, though don't recall exact timings. Wasn't fast, but not annoying and we had budgeted lots of time for this and luggage, since we couldn't check in until lunchtime anyway (Japan is more punctilious about checkin/checkout times than other places, we found). We got a coffee to let rush hour pass, and picked up cashcards at the metro stop. All very easy.
I don't recall the BA meal plan, but didn't feel underfed.
On arrival we headed straight up to Tokyo around Ningyōchō because that was the plan this trip, but on the way back to airport, we discovered that the fast train from Kyoto stops at Shinagawa, where you can pick up the Keikyū Main Line Express to airport...you don't have to go all the way into Tokyo main station and back out again. For next trip, we're likely to head straight out towards Kyoto from the airport, so will use this trick to save a little time and avoid Tokyo main station, which is a very confusing rabbit warren of a place.
Also worth a note, maybe off topic-- the night before our return flight, we stayed at the Villa Fontaine (connected to HND-T3 indoors via a mall) which has an onsen on the roof, with a view of the apron and runways...including an outdoor area. I'm not sure whether it's the best onsen, but certainly quite an experience sitting in the mineral water watching the planes, and meant that we got on the return plane in the right sort of laid-back mood. There's two halves: the Fontaine Grande and the Fontaine Premier and Mrs.Rat (who is in charge of hotel matters) was very insistent on Premier. There's a Yakiniku place in the mall below which was very good and serious value.
On return at Haneda, there's several OneWorld lounges...by far the best was Cathay, and it was very very very quiet, which was just fine.
Enjoy!