I think it depends on what you're doing and what you have time for. It actually sounds like you only have time for one or the other, and that's fine, even preferred! Though remember to do a warmup before weights.
But if you would rather do both on the same day, I personally think that is fine, especially if you are not "lifting heavy" like Ian and also not doing super intense or long cardio. Are you only walking on the treadmill? If that's the case, then it is definitely not a problem at all. Just listen to your body, and if it actually does feel fatigued, then don't do both on the same day.
I usually do a shorter (15-20 min), less intense cardio session before weight training. I only do about 30 min of weight training and I only do it once or twice a week as I'm simply trying to keep the muscles I have, not actively trying to build. It works for me! Do what works for you