The designated lounge at T1 (for domestic/CTA departures) doesn't have showers. I'm not sure if BA have relented to allow domestic passengers to use the international lounge that has showers. If not, you could go through arrivals at T5 use the shower at the arrivals lounge there, then get the Heathrow Express, every 15 minutes, to T1.

But you will want to reflect on how much time you have.