The actual meaning of the colloquial phrase of photons "acquiring particle properties" or "acting like a particle" is really nothing more than saying that photons interact locally and in discrete packages.

Photons can be absorbed by electrons, such as those in light detectors or photoplates; despite the fact that the wave-function of the photon may be distributed across more than one such detector or more than one cell of the plate, we find that the photon is always absorbed at only one location. In the old days of quantum mechanics, one would say that the photon acted like a wave through the slit, and like a particle at the plate.

  What one would say nowadays is that the photon evolved according to the Schrödinger equation until

• it is interrupted by a measurement device, at which point the wavefunction collapses and gives a definite outcome for whatever that measurement device is measuring; or

• until it has interacted in an uncontrolled (but consistent) manner with enough of the world around it that it decoheres, in which case it ends up being in a probabilistic mixture of states which are stable under that interaction.

(Incidentally, the question of "when something counts as a measurement device" is one which is still an open topic at a fundamental level, even though in practical terms we know enough to predict the outcomes of most experiments. A great number of physicists also believe that measurement is in some sense a special case of decoherence. This is all part of understanding the Measurement Problem of quantum mechanics.)

As for the "history" and the "beables" prior to measurement, or prior to the decision of whether to make the measurement or not, these are questions of the interpretation of quantum mechanics. There is no commonly-agreed-upon answer. But the short story is that — no matter how long it took for you to decide whether or not to measure — if you don't measure, the trajectory of the photon is still described by the Schrödinger equation, and you can still cause different "possible paths" to interfere with one another (e.g. in a sum-over-histories description of the evolution of the particle).

The actual meaning of the colloquial phrase of photons "acquiring particle properties" or "acting like a particle" is really nothing more than saying that photons interact locally and in discrete packages, despite being described much of the time by a spatially distributed wave-function.

Photons, when they are left to travel freely, travel as waves. (The same is true of electrons and other matter/antimatter particles.) But photons can be absorbed by electrons, such as those in light detectors or photoplates; and despite the fact that the wave-function of the photon may be distributed across more than one such detector or more than one cell of the plate, we find that the photon is always absorbed at only one location. 

In the old days of quantum mechanics, one would say that the photon "acted like a wave through the slit, and like a particle at the plate". What one would say nowadays is that the photon evolved according to the Schrödinger equation until

• it is interrupted by a measurement device, at which point the wavefunction collapses and gives a definite outcome for whatever that measurement device is measuring; or

• until it has interacted in an uncontrolled (but consistent) manner with enough of the world around it that it decoheres, in which case it ends up being in a probabilistic mixture of states which are stable under that interaction.

This may sound quite similar to the "wave/particle duality" way of saying things, but in practice it gives you a much better shot at understanding how a photon or electron will actually behave when you get your hands on the mathematics.

(Incidentally, the question of "when something counts as a measurement device" is one which is still an open topic at a fundamental level, even though in practical terms we know enough to predict the outcomes of most experiments. A great number of physicists also believe that measurement is in some sense a special case of decoherence. This is all part of understanding the Measurement Problem of quantum mechanics.)

As for the "history" and the "beables" prior to measurement, or prior to the decision of whether to make the measurement or not, these are questions of the interpretation of quantum mechanics. There is no commonly-agreed-upon answer. But the short story is that — no matter how long it took for you to decide whether or not to measure — if you don't measure, the trajectory of the photon is still described by the Schrödinger equation, and you can still cause different "possible paths" to interfere with one another (e.g. in a sum-over-histories description of the evolution of the particle).