Greater surrounding air pressure would cause the water to be pushed out of the buret faster.
But isn't the pressure on the water in the burette the same as the pressure opposing the water flowing out of it? In my mind, the pressures should cancel each other, eliminating any effect from barometric pressure. If there is a pressure differential, how does your particular design cause it?
Temperature change could affect the viscosity of the water, which could affect the flow rate. I haven't researched this, but I think the viscosity change of water is insignificant for normal temperature ranges.
Difference in head can produce significant differences in flow rate. Head is simply the difference in height between the top surface of the water and the metering orifice. It’s important that the head be the same at the start of each measurement. If not, the beginning flow rate will be different and affect the amount of water dispensed.
A water clock has 4 major components.
1. The water reservoir holds water to be dispensed for measuring the time interval. It should have as large a diameter as practical to minimize water level change as water is dispensed. Head can be increased by using a piece of hose/tubing between the reservoir and metering device. This reduces the effect of level change in the reservoir.
2. Control valve. This valve starts and stops the flow of water. It's important that it can be fully opened/closed quickly as any delay directly effects the time measured
3. Metering device. This device consists of a small orifice to restrict the water flow.
4. Measuring device. Measures the amount of water dispensed. A digital scale could be used to weigh the water, but in my mind this would violate any prohibition against electrical/electronic devices. One option is to capture the water in a small diameter clear tube as it is dispensed. The tube can be marked with graduations corresponding to time.
If I recall correctly, a small drop of liquid detergent (or photoflow solution) to break down surface tension of the water makes the clock more consistent.
Here's an example of a simple water clock.
I saw this clock in the hotel hall way back in the late 80s the night before the national tournament. It was made from a couple of boards, 2 1-gallon clear plastic jugs, a garden hose shut-off, a piece of small brass tubing, a long peace of clear tubing, tape and fasteners as needed. It’s been a long time and I don’t have any photos, so some details are not too clear. As is often the case, it was completely rebuilt and re-calibrated in the hotel hall the night before the tournament.
The clock was capable of making repeat measurements of around ¼ second resolution. I believe it actually measured in 10ths of a second, but that resolution was not completely reliable.
One jug, used as the reservoir, had a hole cut in the bottom for easy filling with water. A line marked on it identified how much water should be added. This made the “head” constant so the water pressure was the same at the beginning of each time interval.
The control valve was a garden hose shut-off that turned 90 degrees from full off to full on. (Its operation might be made easier by disassembling it and coating the parts with a lubricant such as Vaseline.) The control valve was attached to the reservoir with black electrical/friction tape.
The metering device was a piece of brass tubing (inside diameter ~ 1/16 in). Electrical/friction tape attached it to the shut-off valve.
The measuring component was a long clear tube wrapped several times around a second 1-gallon jug (The only purpose for the jug was as a form to hold the tubing). One end of the tube was fastened under the metering tube to catch the water as it was dispensed. The other end of the tub was held above the greatest height water reached when timing the longest time interval. Marks added at intervals to the jug or tubing to indicated the time measured. Both ends of the measuring tube were open to the atmosphere at all times.
After measuring a time interval, water was removed by lowering the bottom end of the tube, capturing the water in a plastic cup, and pouring it back into the reservoir.
Accuracy is very dependent on the calibration process. Start/stop signals can be a large fraction of a second long (possibly as much as a whole second). An effort should be made to turn the water on/off as soon as possible after the beginning of the start/stop signal.