If your Mini was financed
...your Market Value motor insurance pay-out could well be less than the amount outstanding on finance at the time of claim. If it is, you'd have to pay any shortfall out of your own pocket before you could even think about financing a new vehicle.
However even if your motor insurance pay-out was sufficient to clear most, or even all of the outstanding finance on the now written-off vehicle, how much would you have left over to put forward towards the cost of your next vehicle without needing stump up money from your own pocket?
If your Mini was bought cash outright
...your Market Value motor insurance pay-out is almost certainly going to be less than the price you bought your vehicle for originally. Without GAP insurance in place, you'll only have the amount paid out by your Motor Insurer with which to replace your vehicle. Whilst in theory this should be sufficient to allow you to buy a vehicle of a similar age and condition as was relevant to the now written-off vehicle, if you want to buy a newer, better or otherwise more expensive vehicle, you'll have to fund the additional cost out of your own pocket.
If you leased your Mini
...the finance house will calculate a settlement figure which is normally a combination of what they believe the vehicle to have been worth at the time of loss plus the sum of rentals that you've not yet paid for the remaining duration of the lease. Your Market Value motor insurance pay-out could well be less than the amount of this settlement figure and although some finance houses allow you to walk away from any shortfall, others require you to pay some of it and others still might well require you to pay all of the shortfall.
Just like with a financed vehicle, if you're liable for any shortfall, you'd have to fund that out of your own pocket before you could even think about leasing another vehicle plus, any initial rental required to start a lease agreement on a new vehicle would have to be found too.