Is there a measurement for tyre profile or height from rim? I have a small child road bike frame that takes 20 inch wheels. It comes with no information from the manufacturer, but from what I've learned there are two 20 inch wheel/tyre standards 406 and 451. 

Based on measuring the fork, 451 wheels will fit but with very little clearance. If it were an adult road bike frame with 700/23c wheels/tyres I wouldn't be concerned but from what I've seen 20 inch tyres tend to be 'fatter' or rounder than their adult sized counterparts.

So - my question - is there a standard to define how high a tyre rises from a rim? If not, is there a calculation/rule of thumb you can use? Or even a crowdsourced resource of end users real world rim/tyre combinations?
If you look at the ETRTO specification of a tire you are thinking of using, e.g. 47-406 (a 20 x 1.9" tire) the total diameter of the mounted tire will be pretty close to the rim diameter plus twice the tire width: 406+2*47 = 500mm ~ 19.7".  (the designation 20" wheel here comes from the diameter of the common 50-406 tire).  The width of the tire will vary depending on the tread, but will be within a few millimeters of the ETRTO width.

While the most common 20" (406) tires are ~2" / 50mm wide (especially on kids bikes), it is possible to find skinny 20" tires but you need to get skinny rims to go with them.  ( Schwalbe has a chart of rim widths http://www.schwalbetires.com/tech_info/tire_dimensions).

As others have mentioned, if you have rim brakes, only one or the other of 406 or 451 will actually work for you and if it isn't a BMX frame, I would guess it is more likely to be the more common size of 406.