Within Terminal 5 itself, the only showers are inside the various British Airways lounges. There are no public showers, and no lounges which offer paid-for access.
If you qualify for lounge access, you'll find showers at:
- T5 Galleries North - inside the lounge, go into the main area of the lounge then turn left, they're about half way down on your left
- T5 Lounges South - part of the Elemis travel spa on the 1st level, ask the Spa staff near the entrance to the CCR to allocate you a shower
- T5 B - inside the lounge, part of the Elemis travel spa within the lounge, again ask the spa staff
To get lounge access, your options are:
- BA Executive Club Silver or Gold
- OneWorld frequent traveller at Emerald or Saphire (but not Ruby)
- Same day OneWorld arrival from long haul Business or First (show inbound boarding pass)
- Same day OneWorld departure in Business or First (show boarding pass)
Assuming you don't have status, that means you need to book one of your legs into Business. Check with your travel agent, it may not be too much to do so at booking time. Otherwise, BA sometimes offer cheap upgrade deals, especially in cases when the flight is looking full down the back. Keep an eye on Manage My Booking to see if it pops up, otherwise ask at checkin for a paid-for upgrade. It isn't available on every flight, and it isn't available every day, but it's worth checking just in case.
Otherwise, you'd need to clear immigration, enter the UK, take a bus to a nearby hotel, and book a Day Use Room at a hotel. You need to be able to enter the UK, and allow several hours to get through, go to the hotel, shower, return, re-clear security etc. Lots of hotels near the airport offer day rooms, cost might work out being similar to upgrading one leg to business when you factor in the bus fare or taxi! (To a certain extent, the further from the airport, the cheaper the day use rate)
In terms of how long it would take if you go to a landside hotel, arriving early morning you'll be hitting immigration just at the same time as a lot of large planes come in, so the EU lines can be slow, and the non-EU lines pretty dire. Gate to landside would probably be 45-60 minutes at that time of day for non-EU. You need to be back through security 35 minutes before. Given your timings, that won't leave enough to get to a hotel on Bath Road (where the cheaper hotels are), check in to a day room, shower, check out and get back to the airport. Your only option would really be the Sofitel, which is directly attached to the Terminal. It's a very nice hotel, but not a very cheap one... I can't see their day room rates on their website, so you'd need to ring them and ask. I've heard rumours that they may also sell access to their gym/fitness centre, which would let you freshen up for a lot less than the cost of a day room, but again no prices on their website so you'd need to ring for that too.