It is very interesting that you haven't lost weight. How long has it been? What is your diet like?
It sounds to me like there are two problems here: 1) You are not giving yourself enough recovery time between workouts, and 2) You have a slight misconception of HIIT.
I think that you are increasing your time (a minute every other workout) at the expense of your workout. HIIT does and will increase your endurance - but that is a by-product. That is something that you will notice when you do moderate intensity workouts and find that you can do them for longer. HIIT was never designed to be something you did for so long, especially at first. There is no reason why you would be able to increase by a minute every other session. That's extremely quick and unnecessary. It is better to increase your intensity than your time.
The way that HIIT works - the science behind it - is that you go as hard as you can. You should not be able to do that for extended periods of time. You should not be thinking in terms of how many minutes you are doing. You should be thinking, for those few seconds - that your life depends on going as hard as you can. Find the upmost boundary of your ability and break past it. That's what I like about HIIT - you're reaching the brink of your human potential.
Science has shown that people who do HIIT lose more weight - but this is NOT because they burn more calories. They burn calories more efficiently from fat, and HIIT increases the metabolism. You don't need to do HIIT for a long time to get this effect. (This is UNLIKE what many people say about low-intensity. Low intensity exercise burns a greater percentage of calories from fat, BUT people burn fewer calories total. That means that they are still burning less calories from fat.)
First, you should take a week off to recover from this high-intensity workout. Remember, the leading experts in fitness recommend taking a week off every 12 weeks, which is often when you see the most progress. But what you're doing is very high intensity, and your body is not responding the way it has previously. You need a break, and you need it now.
After that, I think you should change the way you have been doing it and try something new: cut down your time considerably, maybe even back to 4 minutes; focus not on how long you're doing it, but how hard; go as hard as you can; take quick breaks; and start by doing it only do it for 20 seconds with a very short break, maybe 10 seconds. Try that for one workout, knowing you'll only do it for 4 minutes.
If you want to lengthen your time past four minutes, do it much more slowly than you have been. Focus much more on your intensity. Instead of letting time dictate your workout, your workout should only go as long as you can go at top speed. Forget completely about predetermined time increases. Just do as many repetitions as you can at top speed. When you start to slow down, your workout is over.
In traditional HIIT, when people went below their max capacity, that was the signal to stop. What you say here is a clear signal that your workout is over, and that's ok, as long as you went as hard as you could:
"I couldn't recover in time and the 30 second sprint become more of a 30 second slow run. Eventually the 30 second rest period went from a jog to a 30 second walk. It came to the point where I had to change the ratio to a 20 second sprint and a 40 second rest cycle (combination of jogging and walking)."
Also, don't do it more than three times a week. You may not be allowing your body enough time to recover - which means your next workout will feel lame. Some people add "moderate" workouts during the week and only do HIIT once a week. So 1-3 times a week might be helpful. A moderate workout might be encouraging, as you'll be able to see your progress in endurance.
So again, I think that what's holding you back is your are determining your success by the wrong measures. Your success has nothing to do with how long you can do HIIT. It has much more to do with how much better you can do it.
