The order does not matter, you can see that by focusing on θ=12 which is always equal to 1N2, you can also see it from the conjugation rule where you end with Beta(3,2) no matter the order.
If you wanted the order to matter you could down weight earlier shots or widen the uncertainty between the updates, so previous posterior becomes a slightly wider prior to capture the extra uncertainty from the passage of time.
The order does not matter, you can see that by focusing on θ=12 which is always equal to 1N2, you can also see it from the conjugation rule where you end with Beta(3,2) no matter the order.
If you wanted the order to matter you could down weight earlier shots or widen the uncertainty between the updates, so previous posterior becomes a slightly wider prior to capture the extra uncertainty from the passage of time.