I currently use a mountain bike to commute to work but I am keen to switch to a bike more suited to commuting. I have been looking at road and cyclecross bikes but I am concerned the more horizontal seating position of these sorts of bikes would cause issue with my glasses; specifically, I would end up peering over the top of my lenses rather than through them. Is this a common problem? Do I need to worry?
Yes, in most cases normal prescription glasses sit too low on the nose for the purpose of shielding the eyes in road bike position. And as the speed increases, the need for effective eye protection grows, too. Consider either:
Getting contact lenses, and then you can wear bike specific glasses
Getting bike glasses with prescription lens inserts. In my experience, they are somewhat awkward to handle and not all optometrists will undertake fitting the lenses to what they perceive as nonstandard frames.
Ordinary prescription glasses are sufficient but suboptimal. Although I agree that the best answer is to get bike glasses with prescription lenses or contact lenses, waiting to get optimal eyewear is no reason to avoid using a road bike. I wear prescription glasses and have not yet budgeted for cycling glasses, but I have not had any major problems.
Cycling glasses would make the biggest difference when you are riding above 20 mph. If you normally ride slower than this, or if you don't have the budget, then you can wait until later to get cycling glasses or goggles.
Several months ago, I finally switched to contact lenses, and they have made a huge difference. In particular, the wide field of view afforded by contacts with cycling glasses is much better than what I might hope for in all but the most expensive prescription cycling glasses. I believe that switching to contacts was much easier (and much cheaper) than trying to find prescription cycling goggles with a wide field of view.
I still think it's better to ride than to wait for the perfect solution, but getting contacts and a decent pair of cycling glasses was surprisingly easy. I switched to contacts primarily because of cycling, and I'm glad I made the change.
Eyeglasses, particularly those with relatively "low profile" lenses, will cause problems in an extreme aero crouch position. So it depends on the specifics of your glasses, the bike, and your usual riding position.
I'm extremely nearsighted and wear glasses that are about midway on the round/low-profile axis. I ride a touring bike (ie, a road frame) with a relatively high drop bar. On the hoods I have little trouble with my glasses. On the drops (which I rarely use) I have to tilt my head up a fair amount to see through the lenses, but can when needed. (I can generally see adequately to steer when peeking over the tops of my glasses.)
My bigger problem is seeing through the glasses AND my helmet mirror at the same time. Requires careful positioning of my mirror and some twisting of the head to see behind adequately while on the hoods, and it's a lost cause while on the drops.
I find that using a "Croakie" eyeglass strap helps a lot by keeping my glasses from sliding down my nose.
I've never heard of this being a problem for anyone... but I admittedly stick to the more upright-seated end of the bicycle continuum.
I suggest you try before you buy, of course, but you may also wish to look for a bike with a longer stem, so that if you find you have this trouble, you can raise your handlebars.
Cyclocross bikes should actually be a bit more forgiving of an upright stance than straight-up roadbikes.