You can just buy an Oyster card at any station at either a ticket vending machine or a ticket office. £5 deposit and some cash on the card, any balance left over at the end of the trip can be reclaimed at any TfL ticket office (Transport for London). You can’t reclaim the balance and return the card at a DLR (Docklands Light Railway) ticket office although the Oyster card will work there.
I would use Oyster cards rather than ApplePay as you can only need a single transaction to fund the cards rather than everyone needing ApplePay on their phone with a suitable card. If you are flying in to Heathrow in the morning the prices will be a lot cheaper if you can delay your travel into the city till after 9:30am, you can then get an all day pass for all zones. Gatwick is not reachable with an Oyster card, you need to buy a train ticket, don’t get the Gatwick Express, take a local train it will be a little slower and a lot cheaper. FWIW it worth I usually find myself buying an oyster card every other time I go to London as I have forgotten to bring one with me.
I would go for EE or 3 SIM cards as they will be easier to buy top ups for in stores rather than fighting with their in phone top up via a card. Both have decent LTE/4G networks in the London area and ok pricing. Your phones are unlocked I hope 
It is worth studying a tube map showing the accessibility of the underground as there are some stations that can have you hauling suitcases up and down multiple flights of stairs and long corridors, eg Bank, usually an alternative route can be found that avoids 10 minutes of cardiac exercise in the bowels of London.
HTH
Fearghas