I can't find the underlying legislation to back this up, but the Jersey government's website says:
If you're living in Jersey and you've paid contributions in the last six years, you should receive a letter from us about three months before you reach your 65th birthday.
(The pension age is actually 60 for women who were registered with Social Security before 1975)
The page then goes on to talk about a form you can fill in if you don't get a letter, for example because you live outside Jersey, and explicitly says:
We can't back-date a pension for more than six months before the date you claim.
So I doubt you have any way to get back the lost 2.5 years. They won't consider they had any obligation to write to you. You could try using the Jersey government's complaints system, maybe using your ill-health as an extenuating circumstance for why you couldn't claim earlier, but it seems unlikely you'd get anywhere.
As to where the money that you missed out on would go, Government pension funds don't have a specific pot of money allocated to an individual. Indeed, they can't know how much your pension will cost as it depends on how long you live. Pensions are paid out of the general funds of the scheme and any money not paid out just means the scheme as a whole is in slightly better shape financially and ultimately more likely to remain afloat or require less government support. Many government pension schemes don't have a specific fund at all - pensions are just paid out of the government's current revenue.