Most councils allow you to report missed bin collections on their website. If the council haven't collected due to the weather or staff sickness, then as long as you report it, they will usually empty the bins. I'm assuming you haven't missed any collections due to not putting your bin out in time? Ours have to be out by 7 am on the day of collection. We usually put ours out the night before. I know with Christmas, most councils tend to run behind, so the bin collection day changes from the normal day.
Most areas now have bin collections every 2 weeks - alternating between normal rubbish and recycling. Some areas have food waste bins as well as green bins for garden waste. So your collection frequency isn't so uncommon. Our council will only take the rubbish in the bin, no other bags, that won't fit, again this isn't uncommon.
Buy a storage bin, so keep any extra bags off the ground and contained. Gradually start adding one per week to your normal collection. The alternative is to pay for someone to take the extra rubbish you have to the local tip, if you don't have anyone who can do this for you.
Have you reported to the council that your bin hasn't been emptied? What about other people in your road, presumably if your bin was missed, so were others? You can't be the only person with lots of extra rubbish. If you are, then either other people have taken theirs to the tip or you have missed a collection.