The showers in the Cathay lounge are the best of the lot IMHO. Occasionally there's a waiting list for showers and I have been refused entry to the Cathay lounge once due to "overcrowding" but it's a great place to be.

In terms of other options you can also use the showers in the T5 Arrivals lounge. It's landside, so follow the "Baggage claim/Exit" signs, rather than "Flight Connections" if you want to use this. There are 98 showers there so you shouldn't ever have to wait, albeit they have somewhat of a strange "hospital" vibe. The Arrivals lounge is the best BA lounge at LHR (other than the Concorde Room) in my opinion - it's usually very quiet and has a great selection of food and drink.

It does mean you have to go through passport control, but if you're a US/Canadian/EU etc. citizen this is really just a formality since you can use the e-gates and get through in a few seconds. You can then use the Heathrow Express, Elizabeth line or Underground to make the free transfer to T3 (either collect a blue transfer ticket or use an Oyster/contactless card), rather than the inter-terminal buses.

If you don't fancy the above and you're refused entry to the Cathay lounge, you could alternatively also use the showers in the BA lounge at T3. They're a little more upmarket than the ones in the T5 Arrivals lounge and I've never had to wait for them. The Qantas lounge also has showers, although I've not used them so can't comment.