Manufacturing capability is the main limiting factor
As is the need for a second dose before protection is full-strength, and then a furtherb2 weeks minimum after 2nd shot.
So even though there should be an impact noticeable just a few weeks - remember there's a 2-3 week lag from time of infection to time of hospitalisation, as well as the 2 weeks for first shot to take proper effect. So minimum 5 weeks before a serious infection prevented by shot 1 to be a drop in the figures.
And of course with 12 weeks between shots, and a further 2 weeks for 2nd shot to produce full effect, that means it is 14 weeks from first shot. April is 14 weeks away.
Yes, of course by then there will also be more weeks worth of 1st shot vaccinated people. And depending on the numbers they can do each week, we might be seeing fewer hospitalisation.
If we do 1 million people a week, then category 4 maybe 5 will be receiving first shots by start April (CEV/70 yos, and 65yos).
The sun will be stronger by then as well, so favourable conditions for seasonal drop in cases too