Using RF for relative direction finding for short ranges (sub 10m)

You might want to look in the ARRL Handbook for a variety of practical approaches, especially since you are talking about a signal within the 2m ham band.

Generally, there are three classes of techniques in use:

1) Very crude - "body fade" basically you use your body to block the antenna of a handheld in one direction, and see what that does to the received signal strengh

2) Specially designed antennas, either with a peak or a null in their response pattern. You sometimes see simple yagi-type beam antennas used with wildlife trackers, etc, but iirc nulling designs can actually be made more narrow than peaking ones. Signal-strength based methods tend to be inconclusive when very close to the transmitter.

3) Electrically rotate the antenna at a high rate in order to introduce a doppler shift phase modulation which can be recovered with an FM (or more properly, phase modulation) receiver as an audio tone, the relative phase of which is the bearing to the signal. This can be done with a circle of 4 or more antennas and electric commutation between them in turn with PIN diode switches, or it can even be done with just a pair of antennas, in which case the doppler modulation will null when the antennas are equidistant from the source - something you can detect by just listening to the audio.

Some types of nulling antennas, and also the two-antenna doppler method, are unable to tell the difference between the bearing to the transmitter and its reciprocal. To avoid walking in the opposite of the needed direction, you can move perpendicular to the bearing to take a second reading, or perhaps see if a unidirectional signal strength method (perhaps even the body fade trick) will tell you which of the two possibilities is correct.

I don't know which war you might be thinking of, but in general that kind of direction-finding involves techniques other than TDOA.

The TDOA technique works with the cheap FM walkie-talkies because the antenna switching causes a phase shift in the received signal, which can be decoded by the FM detector. (The two are closely related — phase is simply the integral of frequency.)

The technique requires a "clean" (unmodulated) carrier, because any modulation introduced in the transmitter would also appear at the speaker output of the receiver, confusing the algorithm that's looking for changes that are synchronous with the antenna switching.

Based on a preliminary look at the nRF24L01+ datasheet, it doesn't appear that the same antenna-switching trick will work with this chip. It does all of the protocol processing internally, and simply doesn't provide the low-level access to the signal that you'd need for the TDOA technique. Yes, if anything, you'll simply degrade the communications.

Bonus marks if you can tell me why that guy is switching the antennas four times before taking a reading... is once not enough?

The problem is that everything is AC coupled, and you need multiple cycles in order to allow the transient effects to die out. The antenna switching is driven by a capacitor, which converts the 0-to-5V control signal from the micro into a ±2.5V signal to drive the PIN diodes. Also, the audio path through the receiver is AC-coupled, which means that the output associated static changes in carrier phase dies out over time. For both of these reasons, you need multiple cycles of switching in order to get an unambiguous result.