Indeed no showers in the Domestic T1 lounge (the former London Room), and you won't get access to the International lounge in T1, which does have a few showers. So T5 is your best bet, however note that the Arrivals lounge closes at 14 hrs (and they can get a bit tetchy at 13:45 hrs!). This will engender a landside transfer, which takes a little longer than the airside transfer.