Making a connection at Heathrow, T3 arrival on AA from USA, T5 departure on BA

I am by no means an expert, but if you have a single through-ticket to Basle, I don't think you'll need to retrieve your luggage at Heathrow. It should be checked straight through.

Assuming you do not need to claim and re-check bags, there is an "airside" transfer path between terminals---a dedicated bus for this purpose. Just follow the "flight connections" signs from T3 to T5.. You do not leave the security zone and do not go through immigration/passport control. You do get re-screened by security at T 5. This applies if you have carry-on bags or bags checked through.

The minimum legal connect time for this transfer is 90 minutes and you have more than that, which is good in case your arriving plane is running late or has to wait for a gate. Last time we did this transfer ( in the opposite direction, T5 to T3), I timed it. We took 45 minutes, gate to gate, including the re-screening. But you cannot count on you incoming flight being on time. Lately we have faced delays of up to an hour on our flight from the US, even though it left on time. Often we have to circle 2or 3 times before landing. And once the gate jetway was broken, so they had to bring up a mobile stairway so we ould disembark. That took an extra 45 minutes. So it is Lways good to have a but of extra time---which you do have.

You will not cross the UK border so no passport control or customs.

On arrival you make your way to your departure gate. Any luggage you checked at DFW you will collect at Basel.

So a legal connection time of 90 minutes is perfectly practicable.

http://www.heathrow.com/flight-connections

You do not need to collect your baggage or go through immigration and customs. You will stay airside. Follow the flight connections signs to T5. You will go through a security check where the will check your documents. If you go to the Heathrow website, you can plug in your flights and see the steps in the connection process:

http://www.heathrow.com/flight-connections

By the way, 90 minutes would have been sufficient if your plane landed on time, but many people are uncomfortable with short connections.

Assuming you mean Basel and not Basil (which is a herb) ... :-)
If you have booked this on one through ticket, you luggage will be labelled through to Basel, and you will not see it again until Basel.
As the others say, at Heathrow you will stay "in transit", do an "airside transfer" and never legally enter the UK, This will be faster look for signs as you get off the plane.

For future reference:

Clear Customs by 12:00
Collect baggage and clear Immigration by 1:00 pm

That is the wrong way around, It is Immigration, pick up bags and then customs. That is the order it will happen at Basel (or anywhere else).
Immigration / passport control is where they check whether you are allowed into the country, for how long and stamp your passport. Customs is the check on your goods, whether they are allowed in the country and whether any customs taxes have to be paid. Logically it has to be after you pick up your bags, as they may want to see the contents.
But, in practice, customs does not stop 95%+ of passengers, that would be a waste of time. They just watch you walk past and stop people who look suspicious. It is immigration that takes time.

Final important point: At Basel airport make sure you take the correct exit through Swiss customs (not through French customs). Basel airport straddles the border, and you can exit the terminal into either country. From the Swiss side, bus 50 leaves every 7½ minutes to the city centre.

We did it May on the dreaded, and never-to-done-again, separate bookings. And, we did it in 90 min. Because our second flight was a separate booking, we had to join the sprint towards immigration, collect our luggage, walk through the customs door, catch the free Heathrow Exprss to Terminal 2, check in with Aer Lingus, go through security, and wait.......

Heathrow is exceedingly well marked.

I have done the terminal 3 to terminal 5 transfer several times. If you keep your wit's about you, the transfer usually takes about an hour. You will pass through security and have your picture taken (electronically) and it verified that you are holding a valid passport. You will then take a bus to terminal 5 and pass through security again. Make sure all your electronics are out and functioning and remember to remove liquids, jells and pastes from your carry-on.

If you checked luggage, it will be transferred automatically (assuming both flights were booked together).

A four hour layover is actually a really comfortable time frame. It will give you the opportunity to sit down for a meal and then do some shopping (to pass the time).

One thing to note if you've never flown in/out of Heathrow, they don't announce gate assignments until about an hour before the flight is scheduled to depart. Just keep an eye on the "big boards." As soon as the departure lounge for your flight opens, the gate will be listed and you can head that way - otherwise, just hang out in the main concourse.

The last two times I went through, the bank ATMs in Heathrow served up € at an exchange rate very similar to the bank ATMs I encountered in Germany and Italy. The £ exchange rate was favorable as well...