Heathrow Arrival: General Procedural Questions

I do the "freshen up" on the plane before landing, as do many others in my observation. Part of the waking up process.

On the vast majority of tube stations, if not all, you won't need to remember to tap out as there will be barriers you can't get through without doing so. Note that it's different on the DLR, where the readers are on the platforms and you must remember to search them out. It helps that all the locals will be doing the same so just watch them.
To get to Aldgate East the cheapest way is to take the Piccadilly line and change to the District line at either Hammersmith or Baron's Court - the change is a very simple 20 feet walk across a platform. Cost £5.60 pp (children half price with Oysters with the Young Person Discount added).
Slightly quicker and more comfortable, but more expensive is to take the Elizabeth Line to either Liverpool St or Whitechapel. You can then change to get to Aldgate East but it might be just as quick to walk depending on exactly where you are staying. This is £13. 30pp full price.

I've come in to Terminal 2 the last couple of times. The baggage reclaim hall is huge with high ceilings. Look for signs above you, then look for your flight and it will tell you which belt to go to. Similar to MSP, but MUCH larger. I, too, use the airplane lavatory for my freshen up after breakfast, but I don't take up to 30 minutes. Do you belong to a sky club/lounge where you might go? Some airlines let you buy a one day pass under certain conditions. You should probably repack at a gate out of the way of quickly moving passengers, if possible, but the neck wallets should be checked and put inside your shirts in the stall in the restroom. I've never purchased my Oyster card or added to it at Heathrow, but most Underground stations do have a kiosk where you can do that. Have fun, take a deep breath as you leave immigration and head for your luggage and move as slowly as you need to to recharge. There is usually a good coffee stop in the terminal (sometimes past security) to get a little energy boost. I recommend Costa if you can find one.

I would also suggest doing as much freshening up on the plane before landing. Use your bottled water to brush teeth in the bathroom on board, reapply deodorant, etc. Most international flights I've been on the come through with warm towlettes to refresh your face.

After passport control at Heathrow, if I recall, there are bathrooms nearish the luggage carousels (I've used them before). If it was me though, I'd do the "light" refresh on the plane and wait until I get to my hotel to do a full clothes change. Even if I was a bit stinky (as I generally am after an overnight flight). If you can't check in yet, the hotel should have a bathroom you can use for a clothes change and more comfortable seating for those who are waiting.

And for repacking items, I'd also pull off to near a wall in the luggage collection area and do that there, before entering the hustle and bustle of Heathrow beyond the arrivals area.

1 - immigration
2 - pick up bags
3 - customs
4 - get Oysters

Sorting out your carry ons - up to you really. Probably easiest once you have your main bags.

Freshening up - whenever you see facilities that aren't too busy. Personally I always want to get my bags without delay. We normally have one of us wait at the carousel as the other goes off, then swap roles.

The important part of the Oysters is getting staff to add the discount. You can't not tap in or out on the Underground as you can't get past the barriers. If you are using the DLR at any point you do need to look for pedestal pod things rather than barriers. If you are using buses, tap in only. There's just the one bank of machines to sort the Oysters, there's no secret quiet bit - personally never seen them that busy.

Personally I would be taking the Piccadilly line to Baron's Court then District line to Aldgate East - £5.60 adult fare. 1h15m and a ten foot walk across a platform to make the change. The EL is 51 minutes to 1h06m and will be £14.90 as it's £13.30 plus a single on the Underground, so will hit the zone 1-6 capping. Yes, it's got a wider aisle but you are getting on the Piccadilly line at the start so can choose the first or last carriage and sit near the end of seats.

Chiming in to suggest that a 30 minute refresh session in an airplane bathroom just before landing is going to back up the queue for the loo and that's no bueno for the rest of the passengers who will want to do the same. I'd suggest instead, everyone makes a comfort stop on the plane before landing and then plan to get more cleaned up and situated after you retrieve your luggage and there are larger facilities to do such. You shouldn't have a problem finding a spot to pull over and get situated - just apply general self awareness and you should be fine.

Evaluate whether the half price travel of the Child Visitor Oyster offsets the upfront additional £7 fee for purchase. For a lot of people this will be the case with their travel plans but there has been one person on here for which it did not with lower than expected usage. Don't put a high balance on them, start with £10 and top up as necessary at the start of the day keeping below the £10 to make the refund of the balance easy at a machine at the end.

I hate spending any more time in the airplane bathroom than I have to so I vote for washing up near baggage claim…in shifts like someone else suggested. Pack a washcloth or small towel in a baggy in your carry-on for drying your face along with a travel-size face cleanser…paper towels just don’t cut it for me.