Let $A$, $B$, and $C$ be the ages of Amy, Ben, and Chris as whole numbers on 1st January $2016$, so that
$$30 \le B + 15 \le C \le 99, \quad 15 \le A + 15 \le B \le 89, \quad 0 \le A \le 79.$$
Since $B$ is a factor of $2016$, it must equal one of the following numbers: $16$, $18$, $21$, $24$, $28$, $32$, $36$, $42$, $48$, $56$, $63$, $72$, $84$. Since $B-1$ is also a factor of $2015$, we deduce that $B=32$.
When was the previous lucky year for Ben? Certainly, we need to go no further back than $2016 - 16$, since that is divisible by $32 - 16$. We also need to check that $2000+n$ is not divisible by $16+n$ for $1 \le n \le 4$: $2001$ is not divisible by $17$, $2002$ is not divisible by $18$ (or even by $3$), $2003$ is not divisible by $19$, and $2004$ is not divisible by $20$ (or even by $5$). $5 \le n \le 14$ can also be ruled out, but we do not need to do this because of the "more than a decade earlier" phrase in the problem.
Thus, Ben's last lucky year was indeed $2000$, and $C-16$ is a factor of $2000 = 2^4 \cdot 5^3$. Since also
$$(32+15)-16=31 \le C-16 \le 83=99-16,$$
$C-16$ must equal one of $40$, $50$, and $80$. We want to minimise the age between Amy and Chris, so we must pick the least of these, i.e. $40$. Therefore, $C=56$. Meanwhile, we want to pick $A$ as large as possible so that $2016$ is divisible by $A$, and $2015$ is divisible by $A-1$, and $A \le 17 = 32-15$. The $2016$ constraint means that $A$ is one of $1$, $2$, $4$, $8$, $16$, $3$, $6$, $12$, $9$, $7$, $14$. Examining these, beginning with the largest and continuing until we find $A$ so that $2015$ is divisible by $A-1$, we see that $A$ must equal $14$.
Thus, $C-A = 56-14=42$. To minimise the age difference, we assume that Chris's birthday is on $1$ January, and Amy's is on $2$ January. The difference in their ages is then $41$ years and $1$ day.