Here is a plot of death registration frequencies by age for the UK in 1974.
I see distributions like this quite often: there is some event (e.g. death) which happens either close to birth, or according to some other reasonably well-behaved distribution.
I don't have much exposure to time-to-event/survival analysis/actuarial statistics, but it feels like this must be something that has been explored a dozen times over in a subfield that isn't mine. I've tried searching the literature/Google, but I really don't know what terms I'm supposed to be searching for.
My first inclination would be to model this as a mixture model with two components (perhaps Gaussian, perhaps Poisson). This makes intuitive sense – each unit is drawn from one of two populations with some probability and either experiences the event postnatally or over a longer time horizon. Is this sensible? Is there some other well-established set of models for this I simply haven't come across?