The tenants DO NOT HAVE TO MOVE OUT because you served them a notice.
What you can do now, since they all want to get a new tenancy agreement, is to get them to all sign a surrender to the tenancy. They don't have to sign that, or a new tenancy agreement if they don't want to.
Otherwise you can serve a section 21 notice now, or anytime. Or wait for one of the joint tenants to serve notice as soon as the periodic tenancy starts. Problem with that is that it cannot expire until the end of the second month of the periodic tenancy.
All views posted reflect my personal opinion only and do not consitute professional advice which I am not qualified or knowledgeable enough to provide.
My keyboard also seems to be skipping characters, so apologies if some of my words appear munched
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