There is a pay-per-use lounge at LHR T4 (Holideck) but it does not have showers.

There are showers in one of the T4 toilet areas, if you can manage to find an employee to let you in and give you a towel. I do not know what it costs these days, maybe 4-5 GBP.

As far as I know there is nothing at BOM that fits the bill.