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At a minimum you will want a good set of accurate long nosed calipers to measure diameters. After that a digital indicator for depth guaging would be useful for depths. You can perform a wax molding on backbores and throats to assist with shapes and measurements as well.

Many throats are drilled. The drills used have letters instead of numbers. For example, a "P" drill bit is @ 0.323, which is a common throat size. So if you're really into it, you could purchase a few drill bits in sizes "N" through "R" or "S," which would encompass the range of the vast majority of tuba mouthpieces for all tubas, and carefully fit a drill bit into the throat to determine diameter that way. Or, if you are concerned about scuffing the throat with the cutting edge of a bit, you could purchase some dowel stock and turn down lengths of the dowel stock to the corresponding diameters, which you can find here:

http://en.wikipedia.org/wiki/Drill_bit_sizes" target="_blank" target="_blank

As far as measuring inside cup diameter, there is no "standard." I usually hold a mouthpiece up to a light, and the ring of reflection of the light under the top edge of the rim is where the rim transitions in its contour from convex coming off the rim to concave in the cup, and that's where I measure. Hey, it's as good as any method to measure a complex curve!

iiopes has excellent suggestions. The drill bit method is one of the most accurate ways for throat measurement. An additional safety method I've use is to use Dawn dish soap before sliding the bit in (starting with a smaller bit than you expect and working your way up in size). The soap has been a good, quick, and cheap added level of protection of the silver plating there, especially on some softer brass stocks and thinner plated mouthpieces. Soaking in Dawn or Lemon Joy before using a brush and polish inside really helps break up crud in the backbore of old mouthpieces that have not been cleaned regularly as well.

Considered the shrink?

lost wrote:

imperialbari wrote:Considered the shrink?

More times than I'd care to admit.

Glad you are aware that the tissue mold will shrink when drying.

Klaus