Transactions wouldn't be going through, and the queue would get longer and longer. People would start going elsewhere to make payments, because the Bitcoin network would be taking too long for their transactions to confirm.
That wont happen because you simply have to pay a small fee for your transaction to be accepted. There will not be any queue.
Whenever the demand is greater than the capacity, some transactions will inevitably be delayed. If there is a 60-seat bus every 10 minuets, and 100 passengers arrive at the bus stop every 10 minutes, some passengers will have to wait until the demand goes down to less than 60 per hour. If that excess demand lasts for a few hours, many passengers will have to wait for hours.
It is grammar-school-level math. It does not matter how much the passengers are willing to pay or how cleverly the bus company sets the fees.
The average waiting time will depend only on the network's capacity and how the incoming traffic will vary over time. Fiddling with the transaction priorities will not change the average delay by one second.
Actually we are far below the limit now if you count fee paying transactions.
The average daily traffic now, when there are no "stress tests", is ~120'000 transactions per day, and has grown by 5000 tx/day per month for the last 12 months. The network's capacity, revealed by the stress tests, is ~200'000 tx/day. Since the traffic varies during the day and with the day of the week, congestion will begin to occur well before the average demand reaches the capacity; perhaps already when the traffic will be 160'000 or 180'000 tx/day. Even if the growth is linear rather than exponential, at the curren pace that will happen by mid-2016. (If the growth is exponential, that may happen in early 2016.)
Most of the transactions pay the minimal fee. True, if the minimum fees were raised to a significant level (say 0.20 USD/tx, still much less than the cost of the network), a large fraction of the traffic would disappear, and then the block size limit issue would not be urgent for another couple of years. But neither camp seems willing to do that, for various reasons...