I've flown into both JFK/EWR over the years on a number of airlines. I find the airport experience at EWR marginally more pleasant than JFK, although they both stink in my opinion. NYC area airports really suck. We do lots of things really well, but airports aren't one of them

My preferred way to get to Newark is by the Express Bus (with luggage), or PATH to AirTrain (no luggage):
- Express Bus leaves close to Grand Central every 15 mins, and is $17 one-way/$25 return. Much cheaper than taxi, and delivers you right to the terminal. https://newarkairportexpress.com/Newark/$/
- Info on AirTrain: http://www.panynj.gov/airports/ewr-airtrain.html

For JFK, either taxi with luggage or subway without. Other than that, unfortunately no really good options. I've tried taking the bus to JFK which also leaves from near Grand Central, but I haven't found them too reliable. They only leave when they have enough passengers, unlike the Newark Express Bus. Maybe that has changed, I haven't tried recently.

Can't speak to the BA lounge or Cathay flight experience but for what it's worth, I recently chose to fly out of JFK on my recent flight because of the JFK-HKG flight frequency. I managed to upgrade to Business, so went through the Business Class check-in desk and express security line so it was all a breeze!