That diagram was produced by the BBC after attending a press conference presented by chief scientific adviser Sir Patrick Vallance. As far as I can tell it is a copy of the diagram used by Vallance in the presentation.
So what figures or model was used to produce it? What measures will be in place to prevent the sharp rise in infections and hospitalisations so that we get the slow gentle and limited rise with a later peak?
Where exactly does Vallance say that?
With a rapid high rise, the capacity is almost immediately overwhelmed, so delaying tactics must be used allowing people to be treated before the (delayed) peak. Vallance himself mentions building up "herd immunity".
Again where are you getting this from?
That is what happened in China, before they went on lockdown. And why Italy, Spain, France are in lockdown.
As Vallance stated there is a risk if you do this too early.
There is, that is why you need to plan for what happens as controls are lifted or otherwise released.
The closest to the desired Vallance chart is this from South Korea:
https://www.statista.com/chart/21095/covid-19-infections-in-south-korea/ Which they achieved without lockdown despite the large initial outbreak. It's down to good management, testing and open communications from the govt.