Departing and arriving passengers at Heathrow are segregated, and you're only allowed to enter a terminal you're scheduled to depart from. So, T3 will be out of bounds for you, I'm afraid.
The T1 *A business class lounge does have showers, as does the first class lounge. Strictly speaking, I think you have to be departing on an F flight to get into the F lounge, but as the two are adjacent, it can't hurt to ask. The *A lounges are OK. Functional, not that exciting. The BD number one lounge is more interesting, but if you're on a domestic, you won't get in.