If the priority is to get fed and a shower with the least amount of stress, then the best thing will be to get directly from T3 to T5. Depending on the day, this might be faster going through flight connections the “normal” way via bus and through transit security. You could also go through flight connections and then exit through immigration at T5 and head upstairs and go through security that way…this might be quicker if transfer security is particularly messy in T5 but there’s no guarantee fast track from landside will be a picnic either.

AA arrival lounge will have the best showers and the best food. And might be the best bet depending on whether you get in a bit early. But then you have to make your way to T5 and go through security there.

I just think that getting through security and airside in the terminal you’ll be departing from is less stressful as you’ll be where you need to be and can relax once you’re there. If you eat and shower in T3, you’ll be worried if you’ve got enough time (I would be anyway!)

T5 showers have been open for ages and I’ve never had to wait. You can book one via the QR you use to order food anyway.