Let's say there are currently 437 cases (there will be more in reality), the doubling rate is 3 days and there's a hospitalisation rate of 1 in 500.
Day 0: 437 cases
Day 3: 874
Day 6: 1,748
Day 9: 3,496
Day 12: 6,992
Day 15: 13,984
Day 18: 27,968
Day 21: 55,936
Day 24: 111,872
Day 27: 223,744
Day 30: 447,488
Day 33: 894,976
Day 36: 1,789,952
Day 37: 3,579,904
Day 40: 7,159,808
7,159,808 / 500 = 14,319 hospitalisations per day. So after a bit more than a month you are already at nearly 4 times the number of people being admitted to hospital as was the case at the peak of the last wave. But that was the peak of a wave during lockdown restrictions, and we are looking at this scenario as if there are no restrictions, so we can't be confident about when it's going to peak. It could peak any time up until nearly everyone in the country had been infected. If day 40 isn't the peak, then on day 43 we will be looking at more people having been admitted to hospital in the past three days than was the peak occupancy last time round.
I'm not saying this is going to happen - I am just illustrating how the numbers work.