You need:
1. a debounce circuit on the Btn_in
2. a FSM to control which of the 3 counters you want to stop.
3. Now the spec gets fuzzy.
What is the counter supposed to do? Does it count binary from 0-(2^12-1) and you just extract the bits as [3:0], [7:4], and [11:8]? Do the counter bits for the upper nibbles count as if the lower nibbles were still counting, or do they count at the lower nibble count rate?
If you need to count always as if the 12-bit counter was always running and you are capturing the nibbles, then you should have:
3a. 12-bit counter
3b. 3 4-bit capture registers, to hold the nibble for each 7seg while you wait for next button press.
4. some way to reset the 7seg to try again.