Stupid question, do they account for speed dilation on the beam experiment? The difference in lifetime will translate to 50k km/s speed of neutrons or roughly 1/6c
From skimming the review paper from OP (the "source:" in the caption on that error-bar chart), the neutrons in the beam experiments are thermalized, to a mean velocity of ~2,200 m/s. So, slower than 1e-5 c.
(Thermal meaning the neutrons scatter lots of times against atoms in a solid material, until they reach thermal equilibrium. ~km/s is a typical Boltzmann velocity for atom-size things at room temperature).
More technical details. For the time dilation you must correct the time by 1/sqrt(1-(v/c)²) that can be approximate for small speeds as 1+½(v/c)².
If the speed of the neutrons is less than 1e-5 c, the correction is less than 1e-10.
The lifetime of neutrons is approximately 15 minutes lifetime, so the correction is less than 1e-7 seconds. But they are measuring with a precision of only only a few tenths of a second, so the corrections is negligible.
How does the scattering affect the neutrons? When do we start the clock for their lifetime anyway? It sounds like they could absorb + reemit sometimes when being scattered.
It's one of the nice properties of exponential decay. They are measuring the mean time of life (~15 minutes) but it's easier to explain with half life (~10 minutes).
If you have a bunch of neutrons and put them in a box, and look again 10 minutes later, you will see that you have only half of them.
If pick all the neutrons that survived for 3 minutes, and put them in a box, and look again 10 minutes later, you will see that you have only half of the neutrons that survived for 3 minutes.
If pick all the neutrons that survived for 7 minutes, and put them in a box, and look again 10 minutes later, you will see that you have only half of the neutrons that survived for 7 minutes.
If pick all the neutrons that survived for 42 minutes, and put them in a box, and look again 10 minutes later, you will see that you have only half of the neutrons that survived for 42 minutes.
When the time is too long, you need to create a really big number of neutrons initially, so enough survive until you start the experiment.
So ... the waiting time until the experiment start doesn't matter.
It's easier to understand with a discrete model with coins. You have perfectly balanced coins with 50% chance of head and 50% chance of tail. Each minute you flip all the coins at the same time and remove all the "heads". So you can repeat this, and each time you have less coins. You start the experiment, flip the coins 10 times (and remove the heads), and you get 1000 coins that survived. How many additional times should you flip them to remove half of them and have only 500?
It's very unintuitive. When I read your question said: "It's a difficult question, I wonder what they are doing."
These are thermalized neutrons, that means they have bounced a few times in random directions. You can model this and have some kind of convolution of the result, and then deconvolute the experimental data or fit it. But they are trying to measure 1/100 of seconds, and this is possible but very noisy, so it's strange. Wait a minute, it's a exponential decay ... so it doesn't matter ...
It doesn't matter when you start the clock, just that you count at the beginning and end. I doubt scattering plays a big role here at these kinetic energies.
https://doi.org/10.3390/atoms6040070
(Thermal meaning the neutrons scatter lots of times against atoms in a solid material, until they reach thermal equilibrium. ~km/s is a typical Boltzmann velocity for atom-size things at room temperature).
(Not a domain expert).