score:8
Based on advice from FlyerTalk, which suggested AA then CX then BA, I decided to try out the first two. (I know the BA ones from T5). It's a tough job, but those showers won't review themselves...
The short answer - AA, by a nose
The American Airlines lounges showers seem to be located between the two lounges, and I think everyone can have access. (Those in the Flagship lounge get a special keycard to get back in). The room is large and square, with hangers, a decent sized bench off the floor, and a large shower raised up with a door. Looks functional but well done, and no risk of getting your stuff wet! Good water pressure and decent temperature range
The Cathay Pacific (CX) showers are located on the corridor between the lounges, key from reception. The room is large and rectangular, with the walk-in shower at one end, and a nice stone-motif wall. Maybe a tiny bit bigger than the AA one. However, it's a walk-in shower with a partial glass screen, slightly recessed into the floor, so the area and floor near the shower can get wet. Mind you put your clothes nearest the door! Decent water pressure, and a choice of handheld or rainfall shower heads, but the water wouldn't go quite as hot as I'd like. Little bit snazzier than AA, but slightly less functional.
As mentioned in these two threads from FlyerTalk, the BA showers are in the Elemis spa area between the lounges, and are the same model as those in T5, but in better condition (not hard!). The T5 showers were OK but not amazing when new, with good water pressure and jet options, but a risk of water on the floor, and very strange gurgles / noises. These days, the T5 showers are in a shocking condition, with no news on when they'll be refurbished. Even if the T3 ones were almost brand new, they wouldn't be quite so good as the CX or AA ones, and with them being somewhat old (but less worn than T5) they'd be less good.
(No photos as I didn't have a camera that worked well in a space that size)
So, if you want the "best" by my definition, use the American Airlines ones, but if you like a slightly swisher look with a small risk of things getting wet, go for Cathay!