How do I reduce urine odor from a bedroom?

I too think if the odor is that strong Mom maybe the problem. UTIs have a strong odor. Try vinegar in a bowl and see if it helps. When I used it I saw a difference within 24 hours.

And checking everything is a good idea.

It has to be coming from somewhere, this odor, so I think it is still a matter of searching the source. Is it in wood flooring cracks? In the mattress despite a protector? Until you find the source you can't address the problem.

In addition to the great ideas already given, it may be helpful to check areas of the room that you wouldn't ordinarily expect urine, like in the closet, a corner, behind a door, in a laundry basket, in a trash can, etc. My mom is the master of using clean laundry such as a towel or sweatshirt to pee on, then replacing it on the shelf or tossing into the closet. She also is prone to using anything that looks like a chair as a toilet, then covering it up with blankets or stacks of folded clothes. I especially agree with not using a diaper pail...dispose of incontinence products immediately.

cwillie, right about the odor not being inevitable.
A yeast infection, UTI, dehydration and possibly undiagnosed diabetes these are some other things that can cause a strong odor to the urine.

I have to disagree that a stench is inevitable, my house and mom's room did not reek of urine.

Aside from the things Grandma1954 has mentioned very strong odour could be due to infection, you might want to check for that. Not being properly hydrated can also lead to very concentrated urine.

Remove all used incontinence products from the house immediately and do not use 'diaper pails' at all. They hold onto odors. No matter what you do or don't do, however, urine odors linger in a room where an incontinent person dwells. My mother's room had a hardwood floor, no carpets anywhere, she was dry all the time due to frequent changes, etc, and still her room reeked of urine. People say to put a bowl of vinegar in the corners of the room which is supposed to help. Activated charcoal bags you can buy on Amazon also absorb odors; room sprays and perhaps even an air ionizer might help. Look on Amazon for one according to the square footage of the room in question.

Good luck

What else is in the room?
Any carpet?
Any furniture that is fabric?
Any clothing that has not been thoroughly washed?
And did the mattress become soiled prior to the use of the mattress protector? Is the mattress protector waterproof? And has the mattress protector been washed recently? While it keep the mattress "clean" the protector itself becomes soiled if a brief (aka "diaper") leaks. The waterproof ones are a bear to wash since they really should not be put in the dryer so get a second one as a spare.
(alternative would be get a Hospital bed with a mattress that is cleanable and non absorbent)
A small room size air purifier should work to remove odors.

Oh, one more thing that is in the room that may hold urine odor.
MOM.
Make sure that her skin is clean and dry. Use a very light amount of barrier cream if you are using it. A thick layer can make it difficult to clean the skin well. Also make sure her hands are wiped as well. If she tends to fiddle with the brief her hands can get urine on them. So hands and as best as you can the nail beds