You can use a program known as OpenSCAD with the threads library to assist your objective.
The important aspects of a thread are the major diameter (outside diameter at the thread surface), the length of the bolt/nut and for your project, the start count. The noted library has a parameter called n-starts, which covers your triple start requirement.
It may require some trial and error processing, but you have a good foundation with OpenSCAD library.
Even if the bottle is manufactured with Imperial reference (unlikely), you'd probably have an easier time with metric threads. I just checked a Dasani water bottle, which measured at 27.27 mm major diameter, 10 mm length, thread pitch 2 mm. The thread pitch in metric is the distance between peaks (or valleys) from one to the next. This particular bottle has an interrupted thread, but that should not be a factor for creating an adapter.
This Dasani bottle does not have a multi-start thread. It's not a complete thread, but it is single start. Considering that a water bottle top has minimal length, you may also have an interrupted rather than a multi-start thread.
The code for the measured thread in OpenSCAD is relatively simple.
thread_dia = 27.27;
thread_pitch = 2;
thread_length = 10;
metric_thread(thread_dia, thread_pitch, thread_length, n_starts = 3);
The above code would generate a 3-start thread in the form of a bolt. Using a difference() code would provide for the inverse required for your proposed adapter.
top image is single start, bottom is triple start.
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