Honestly latex reactions are usually severe and appear very early. Latex is in tons and TONS of things, so it's really hard to get to adulthood without realizing you're allergic to it. You might be still be sensitive to the latex, but you're more likely allergic to something else in the condom than the latex itself.
It seems odd but a person CAN be allergic to lubricants (even water based ones) and spermicides. I am. I cannot use 3 different brands of condoms because they make me swollen, red, itchy, feeling "dry", and cause pain.
If you used a new type of condom, don't use that type again. Go back to whatever you used before. If this was your first time having sex with a condom you might need to get small boxes and try different brands. There are non latex condoms so it's worth it to try those too. I know one of my male friends prefers to use them because of texture and sensation. (Although he admitted they break easily if there isn't enough lubrication.)
It's also possible to get this feeling if you didn't use enough lubrication. Basically it's the equivalent of a rug burn inside your vagina. (Swelling, redness, pain) If you think that perhaps there wasn't enough lubrication, maybe you should look into a good water based lubricate for next time.
Now swelling and pain during urination can also be signs of an infection. In my experience any allergic reaction or "rug burn" goes away after a day or three- it will slowly get better until you can't tell at all. (Although it needs complete rest for this, trust me you don't want to have sex if the inside of your vagina is red and angry; Pain beyond pain.) If it doesn't go away after a few days, gets worse, or you start to notice lots of unusual discharge (colored, smelly, thickness, clumpy) it might be an infection. If this happens I suggest you see your gyn and get checked out.
Yes, you could be having an allergic reaction. There are other types of condoms to choose from. Try one of those and if you are still having the same problem see your Dr.