Do you have 3d printer and a laser pointer?
You can scan your board with a scanner or a photo to get preliminary coordinates. Then you can use a laser mounted to the carriage to confirm and fine tune your coordinates.
You can also use a camera mounted to the carriage too. Similar to pick and place machines that look down at fiducials and pads.
I can probably make a number of suggestions, but I need some information to pare the response down to something reasonable.
Is this a board you have Gerbers (or CAD source data) for?
How large is the board?
Through hole or surface mount? If surface mount, what pitch?
Discrete stand-alone test point pads, or component pins as test points? If discrete pads, how big and what shape?
how accurate are the board edges (scored or snap tabs?)
How closely spaced are the two closest test points
Test points on one side. or top and bottom?
Does the board have fiducials?
As an example, for modest size through hole boards I don’t have data on, I will usually do a high resolution scan on a flat bed scanner. It’s usually good enough if you use raw uncompressed data and an image viewer that will give you cursor coordinates in pixels or mm.
I’d second that suggestion of finding a CMM service.
Both of the following are assuming those 5 vias around each device are the test points you’re speaking of.
As a mildly “out of left field” alternative, Try this:
Try photo-copying the PCB. That should make a reasonably undistorted image of the PCB that you can then start measuring from. (You might need to de-solder that 330uF cap in the lower right.)
Pick some reference edges, (I’d recommend the top and right edges from the photo you shared, as they are straight, perpendicular, and unblemished.) and have at it with a square.
Pop those dimensions into KiCAD and place the test points.
Make a 1:1 test print and double check your work. Adjust the values and repeat the check.
Once you’re good to go, then order a PCB for that.
Alternatively, if those vias are not filled, what you could do is use them as a template and transfer the locations to something. I’m envisioning that you use a drill (if you can find one small enough) or a dowel or needle that will fit. Ideally, maybe use a piece of ESD delrin for the substrate. You can mark it directly, and then use the marks to come back around with a mill and drill the hole for the pogo-pin sleeve. Finally, use some sort of find adjustment to set your edges.
I would suggest perhaps taking a good digital photo using a long focal length lens, a tripod and good lighting.
Measure an X and a Y reference on the board as accurately as possible (drop in on a machine shop, they will have nice ways to measure).
Then use a graphics program to import and scale the image.
Then just put the holes where you want them.
Assuming (and you know what happens then) the through holes on the traces between ICs are the test points this should be pretty straightforward.
Based on the size of the hand (assuming you aren’t Andre the Giant) the board should be small enough to do a simple bed of mails without specialized locating features or floating pins.
You’ve got three locating holes for positioning the board. There are also optical fiducials near the holes.
The test points (if I have the right of it) are both large enough and open. A pyramidal probe will tend to self center.
I’d plop the board on a flatbed scanner and get the highest resolution raw TIFF output available. You’ve got fiducials near the locating pins, so you almost certainly have some on the bottom. You can use the fiducials or holes as references and distortion checks.
Import the image into your favorite mechanical CAD package and ID the test point locations. You can correct for distortions once you’ve measured the references.
That’s some pretty impressive roasterization on the lower left.
If you used Altium or Autodesk Eagle you can simply push the PCB layout to Autodesk Fusion 360 to create a precise CAD design for your bed of nails.
No photocopier necessary, but you may need some cash for the Fusion 360. (Surely there’s a 30-day trial available? You just need a way to convert the info.)