3 weeks should be enough time to stabilize vision to get a good sense of the outcome. Although some people see a shift after that, its likely not much if it does happen. Mine was the same refraction from the 1 week postop onward, up until a check a couple of weeks ago with only a slight shift that might even merely be due to differences in doctor doing the refraction or the equipment used (e.g. lenses aren't to perfect tolerance exactly the specific measurement).
You mention wearing glasses in between surgeries, but that might potentially be a problem, many people need to wear a contact lens on their unoperated eye between surgeries. I'll guess that may mean you typically wear glasses, and its a reference to perhaps not needing them after surgery on both eyes. The question is how strong a prescription you have now for your glasses. After surgery the your eye will likely either not need correction for distance, or only minor correction. People who were have strong prescriptions before surgery then have a major difference between the prescriptions for their two eyes, e.g. 0D vs. -6D, which can cause problems with trying to wear glasses since glasses don't just change the focal point, they also change the size of the image (magnifying it or reducing it, depending on whether you are nearsighted or farsighted). The amount the image size changes depends on the lens power. If both eyes need around the same lens power, the image sizes are close enough that the brain can combine them without a problem. However if the difference is e.g. over 3 diopters or so, most people can't deal with that much difference in size. If your presription is lower than that then you can likely deal with glasses in between. There are special glasses that attempt to compensate, though I'm unsure how how much of a difference they can compensate for. The magnification effect is less the closer the lens is to the eye, so contact lenses tend to work ok. (btw, some folks who wear glasses haven't tried contacts recently, they are much better than they used to be a couple of decades ago in terms of comfort).
Usually they actually don't tend to prescribe glasses until after vision stabilizes in the operated eye, but if your prescription is low, it'd probably be a safe bet to either remove the lens on the side with the operated eye or perhaps get an optician to replace the lens with a 0D non-prescription lens for esthetics.
If you are going to wear a contact lens, if you don't now, you might just go into whatever the cheapest optometrist is to get trial ones before your surgery. Usually they require you to be out of contact lenses for a few days before a pre-op visit to get the eye measurements exact, so if you were going to wear a contact lens on the other eye you might try asking if they can do the pre-op measurements for both eyes before the first surgery.
Some of the argument for waiting seems to be "but what if something goes wrong in the first eye". However if someone needs surgery in both eyes, even if something went wrong in one eye, the other eye would still need to be done, and of course in this case 3 weeks is more than enough time to know if there is anything major going on. In my case even doing the 2nd eye the next day allowed a postop check to see that I already had good distance vision (and near that was somewhat good, it fluctuated a bit for the first couple of days, going in and out and then seemed to stabilize and seemed to remain the same).
There is actually a rise in the number of surgeons outside the US doing both eyes on the same day since there are some benefits to neuroadaptation to getting both eyes done at the same time, in addition to the convenience of only going through one period of doing eye drops and postop visits. Apparently the major reason it isn't as common in the US relates to lower insurane compensation for that approach, rather than an explicit consideration of its pros and cons. In places where that is done they use seperate operating rooms and equipment to ensure if e.g. there is some source of infection or an equipment problem in one room, that it doesn't impact the other eye.
I'm assuming they must not require bandages afterwards, in my case I wore a bandage after surgery over each eye until the postop the next morning, but I've seen people posting indicating in their surgeries even on just one eye at a time they didn't have a bandage afterwards, so I'm wondering if it might be an old protocol established before modern surgical techniques and it isn't really needed.