It's unlikely the wooden cage would cause such symptoms unless you used some paint (varnish) or glue which may be toxic to them?
You wrote that it readily accepts water. Do you put water in it's snout or do you leave water in it's enclosure? If you put water in it's snout, you may be giving it too much at one time. If so you have to consider that the water may be bubbling from it's nostrils as aspiration may occur. Sort of like you drinking a coke too fast which might exit through your nostrils. If you leave standing water in it's inclosure, you must regularly check the humidity levels in the enclosure. The standing water is not the problem but the often higher humidity levels it could cause might be. Uromastyx are dessert creatures and some may never come in contact with any water (ie, rain, rivers etc). Just keep in mind that higher humidity levels may cause a uri (upper respiratory infection) which show the symptoms of a person having a head/chest cold. This can effect breathing, lungs and also produce mucus. If you see mucus in it's nostrils or throat, it may just be a uri. If the fluid you see bubbling from it's nostrils is clear and watery, It may just be that it's aspirating. If you feel water is a must in it's enclosure, you should only allow them water for a short time then take the bowl away after an hour or two. Whenever I offer water, I usually do this about once a week if I feel it's necessary.
I'd play it safe and take it to the vet for a check up. If it is a uri, they can start it on antibiotics. You should also isolate it from other cage mates. Keep it quarantined till these symptom clear up. Also be sure to clean (sanitize) it's cage prior to putting other uros in it or before putting it back.
John