Getting to and from Heathrow
The Heathrow Connect and Heathrow Express services will operate as usual. Neither do usually accept Oyster cards, but it might be worth checking on the day. They both terminate at Paddington station; the Connect stops at several stations along the way. Travellers could get the Connect to Hayes and Harlington, then leave the train and touch in with their Oyster. All trains (including the Connect) accept those form that station, it seems (but you will have to leave the train to touch in).
By bus or coach
National Express offers connections to Central London (e.g. Victoria station), and you can take Transport for London (TfL) buses (make sure to untick the Tube option on the Journey Planer options page). They will take a long time, and you will have to change a few times (Hounslow, White City or Hammersmith, most likely), but they will get you there. Plus they take the Oyster card, and there are Night Buses (see legend/map keys). See this map for travel to other destinations. (Cf. Terminal 5 bus stands map.)
Other means of transport
You can get a black cab (expensive, especially from/to LHR) or a taxi (usually cheaper, but ask), or even rent a car. Heathrow Airport has more information. (Update: Actually, taxis in London are synonymous to black cabs, while minicabs are "private hire vehicles" - which I - but possibly nobody else here - call "taxi", probably because that's the word I'd use in German or French. Black cabs can be hailed (or flagged down), the drivers are very knowledgeable and they are generally more regulated. Hence the higher price. Minicabs are booked. So, to summarise, a black cab is a taxi, and a minicab isn't, but will still get you home or to the airport. But you could always get an Uber car if unsure, I suppose.)
Transport around the airport
Transport within the Heathrow Free Travel Zone is - you guessed it - free on certain local buses and trains (but not the tube).
Easybus connects London to Heathrow, Luton, Stansted and Gatwick. This is your best bet for LHR-Shepherd's Bush, by the way. Remember, National Rail in Greater London, TfL Overground, DLR, Tramlink, Buses (24/7), Riverboats and the Airline (cable car) should all operate and should accept Oyster cards (although not all accept weekly Travelcards, I think), so once you're in London, getting around outside of the rush hour shouldn't be a problem. To the contrary, you'll actually be able to see the city much better that way. :)
If you have a UK address, you could also join a car club (see the page's right-hand column), but don't expect to find a car during a tube strike... ;-)
Ride by bike (for the adventurous and daring)
TfL Santander Cycles (previously sponsored by Barclays, and locally known as Boris Bikes, after the current Mayor under whom they were introduced) are available throughout travel zone 1 and parts of travel zone 2, and they should be available during a strike, but I have had mixed feelings about them: They are fun to get through the Royal Parks (although not all paths are open to cyclists) and on side roads a bit away from the busy city centre. In the touristy bits of London, though, there aren't many dedicated cycle routes, and you'll often have to share a lane with buses and taxis. Buses, by their nature, are on a schedule, so they will overtake you, just to pull in right in front of you to stop at the next bus stop. And being chased by a bus makes for a somewhat unpleasant experience. On the roads, people will try to overtake you, especially at traffic lights, no matter if there is enough space or not, and often force you dangerously close to the curb, if you travel on the side of your lane. It's therefore super important to ride in the middle of the lane, contrary to what you are used to from home. But even then, I was once nearly run over twice in the span of fifteen minutes. That's why I now consider Boris Bikes the fastest way to your local A&E. They do only take payment cards, but no Oyster.
Also remember that people drive on the "wrong" - as opposed to the right - side of the road. And unless your suitcase has four wheels, taking it along might prove a tiny bit difficult. ;-)
How to pay
Buses no longer accept cash. You can pay using an Oyster card (which is sold at all tube stations and some news agents) or by contactless payment card. Bear in mind, though, that your card issuer's transaction fee might well exceed the price of the actual ticket if you have a foreign card. You can load travelcards or cash on your Oyster, and daily and weekly (Monday to Friday, I think) capping is in place. Payment cards have daily and rolling weekly (7 days) capping, I think. And if you have a travel card for any zone you get buses free in all of them, if I remember correctly.
Oyster cards "cost" a 5 pound deposit, which you can claim back when invalidating them at a ticket machine (e.g. at the airport, when you leave), along with any unused cash loaded onto the card. Just make sure what remains on it is less than a tenner; otherwise you'll have to go to a ticket counter, fill out a form, and provide proof of address and maybe even a local bank account (don't know for sure about the last point, to be honest) and it gets very tedious.
Anyway, always keep the card you want to pay with well away from all other contactless cards in your wallet, or you may end up paying more than once. We call it card clash.
Bookmark TfL.gov.uk on your mobile, get a local prepaid/pay-as-you-go SIM card with lots of data (e.g. Three's All in One 15), and install an app that will tell you which bus stop you need (like the excellent Buses Due for Windows Phone).
Avoid the rush hour
Avoid Angel like hell. Actually, just avoid central London, especially the northern bit of the Northern Line, during rush hour (worst in the morning, I believe), if you can - which is even more true should huge parts of the network be suspended. (Full article.) Unless you're curious what sardines feel like in a tin box, that is...
Don't panic; just breath
Most Tube strikes are usually called off at the last minute anyway... ;)
Also monitor the TfL website on the day; there might be contingency plans in place to keep parts of the network open.
Last but not least
You can potentially claim refunds when you are delayed (and paying by Oyster); see TfL's website for details.
UPDATE: Now Panic and Freak Out! ;-)
Looks like the strike will go ahead, with tube services slowly winding down from 6PM on Wednesday - and no contingency plans. So it'll be a total shut-down of the tube network this time. Yay. Not. :(
The latest updates can be found here.
There will be no First Great Western services between Paddington and
Greenford for 48 hours from 18:00 on Wednesday 8 July as a result of
separate strike action.
An hourly coach service will run from Victoria Coach station to
Heathrow Central Bus Station and Terminals 4 and 5.