There are showers in the cabanas in the Concorde Room which can be pre-booked via You First as per the advice above.

There are also standard shower rooms adjacent to the Elemis Travel Spa between the Concorde Room and Galleries First lounge (also shared with the Galleries Club South lounge above), within the Galleries Club North lounge, and over in the Galleries Club T5B lounge.

There's no need to go landside to use the Galleries Arrivals lounge, as you can use the superior Concorde Room as a connecting passenger, and either get a cabana or use the showers next door.