Not surprisingly, this is actually extremely complicated and only has so much to do with Glycogen. Just to be clear, Glycogen is simply stored Glucose which is kept in your liver and muscles. During exercise your body will use both Free Glucose from your blood as well as Glucose produced by breaking down Glycogen. The actual chemical that is relevant is ATP (Adenosine TriPhosphate), which is literally the means of energy for your cells. Think of it as gasoline you put in your truck. Now, when you are exercising in an aerobic capacity your body uses a combination of Oxygen, Glucose and Triglycerides to produce ATP. If you are Anaerobic, your body essentially has to use only Glucose and Triglycerides. When your body is Aerobic the ATP production is extremely efficient, when you are Anaerobic it is not.
To put it all together, it really depends on what you are doing for "cardio" before you do weights. If you are walking for 20 minutes or doing a light jog, you don't even come close to burning your Glucose and Glycogen stores. If you do a serious bout of HIIT training for 20-30 minutes, then you might burn through a significant amount of your Glucose and stored Glycogen leaving not enough for weights. That will essentially put you in a position where your body is overly fatigued to get good muscle contraction needed for weight training.
Very long story short, I would never recommend doing any cardio before you perform weight training. Typically, you will want to start your workout with a 7-10 minute warm-up. Weights should then never exceed 60 minutes. If you want, you can crank out a quick 20 minute cardio session, ideally something like a HIIT workout, after you finish weight training. Important to note, when you workout you will typically see a big rise in your Cortisol (Stress Hormone) levels after about 45 minutes to an hour. I don't want to nerd out too hard and make this overly confusing, just know the Cortisol spike that occurs is bad. You typically want to keep your workout (not including warm-up) to less than an hour but no more than 90 minutes. For that reason, it is typically ideal to split your weight training days and your cardio days. I know the OP said he only has a couple days to workout, so you can always hit the gym for weights then do cardio sessions at home (i.e. running outside, bleachers and track at a high school, etc.).
Hope that wasn't overly confusing and helps. Let me know if I need to clarify anything further. Tried to keep this short and sweet.