I have a cornet and would like to purchase a mute for it. However, the majority of mutes are trumpet mutes. Will a trumpet mute work properly in a cornet? My main reason of concern is that the trumpet has a cylindrical bore while the cornet has a conical bore. Should this be a concern?
Conical vs. cylindrical bore instruments should not be a concern. The bell sections of all brass instruments are conical and the valve sections of all brass instruments are cylindrical--the differences between them are more subtle and occur elsewhere in the instrument.
You really just need to try the mute on your cornet. Even within the same family of instrument, different designs work with some mutes better than others. The one example that comes to mind is the Martin Committee trumpet. The bell on this instrument has a wider throat than many other trumpets, so using a harmon mute with it requires that you add a layer of cork to the outside of the mute so that it seals properly.
And by that same token, a mute can always be adjusted to fit your particular instrument. Most cornet bells are very similar to trumpet bells, so you'll probably find that the trumpet mute fits your cornet just fine, but it may be better in tune with the addition or subtraction of some of the cork on the mute. In fact, some mute manufacturers (Tom Crown is one) ship their mutes with LOTS of cork, expecting them to be filed down to fit the instrument it will be used with.
No you'll be fine. I don't think I've ever found a trumpet mute to be un-usable in a cornet. You may find that you prefer the sound and feel of different mutes in different instruments though. This of course will vary depending on the instrument and the player.