I called the LH Card hotline a couple of days ago and was told that the VISA - and also the VISA Partner Card if you use one - remain(s) active until you pay with the MC for the first time.
You also keep collecting miles with the VISA.

Once you pay with the MC for the first time, the VISA and the partner card will both be closed automatically and only then will the new MC partner card be issued and sent out.

So you have the choice as long as the VISA is valid.