Is there anyway to tell if a glass bottle of beer is carbonated without opening it?

I combine the bubbles and yeast idea. Observation is a scientific approach. If you turn it upside down quickly and make some bubbles , there is definitely co2 released by the yeast. Oxygen alone may produce a little amount of bubles, but co2 bubles rushes up for a few seconds. You may notice a bit of this activity after the 1 week and it should be doubled in week 2. At the same time, even when the yeast at the bottom may only means the end of it's suspension it also means that the sugar frenzy was over and now yeast rest in peace. Also be patient and wait 2-3 weeks,that is part of the craftsmanship of making beer. Also, take notes of all the steps you've made during the process so you can share in a forum what you did exactly and by looking at the later you may also figure out what went wrong.

This is possible, but not in a scientifically measurable way.

Try this:

1. Hold one of your bottles of beer up to the light so you can see the air gap that expands from the top of the bottle down to the top of the beer in the bottle.
2. Quickly turn the bottle upside down then back again, with a slight amount of force, but no need to shake it.
3. Observe the air gap in the bottle and notice the amount, if any, of bubbles forming at the top of the beer. If the beer is lighter in color, also notice the bubbles floating to the top.
4. Through trial and error, I have found this method to be fairly accurate in predicting the level of carbonation in a bottle, but as I said, it's not something you can measure.

You can compare store bought beer bottles to get an idea of what to look for, but keep in mind that different styles yield different levels of carbonation.

When I brew, I usually will do a number of mini bottles. Like the OP, I'm not one for patience. :) This allows me to have test bottles with minimal waste.

I often open one after a few days, and then I'll often open one every day or two after that. I find it interesting to see how it changes in the 2-3 weeks while it's carbonating. How many bottles are in the batch? Is it a big deal if you lose a few before they were completely ready?

Another thing you can do is determine the carbonation level roughly by the noise and feel of removing the cap. If it seems under-carbonated, just immediately put on a new cap. That bottle will end up slightly less carbonated than it would have been if you'd left it alone, but the difference will be tiny.

I usually wait a couple weeks then look for trub in the bottom of the bottle. If it's there, the yeast finished the job.

No.
If you are that dire for things to happen quickly fill one plastic bottle each time you bottle. It will be firm once the beer is carbed. You can simply re-use 16-20oz soda bottles for this, or buy plastic beer bottles with screw on caps.

A time test is your best bet.

Bottle carbonation should be pretty mechanical unless you're experimenting with conditioning yeasts. If you're not doing anything fancy with the carbonation, it should be safe to assume that your beer is done carbonating after two weeks.