Thanks for your question. I will try to help.
If the notice you have served is valid and the tenant does not leave, then you could not simply wait until they were out and then change the locks. You would have to apply to court for an order for possession to evict them, this is probably what they mean by saying that they could make it difficult for you because it can take a few weeks to get a court order.
However, if you had to apply for a court order then you could also apply for a money judgement against them for the additional rent for the period of time that they have stayed and also your direct losses resulting from them staying. I would make this point to them and state that you would certainly apply for a county court money judgement which would show on their credit file and make it extremely difficult for them to rent a property or get a mortgage offer in order to purchase the property.
I would also make the point in your letter to them that they are contractually obligated to leave the property in a good condition and if they do not then you will simply claim against their deposit and if the deposit is not enough then add whatever additional costs there are on to the amount you will sue them for. You should also state that you are aware they have already breached the tenancy by having occupant not permitted by the tenancy and that you reserve the right to recover losses in this respect as well.
If you have already signed a new tenancy agreement with the new tenant then this is the risk that you run if the current tenant become difficult. The new tenant could sue you for the loss they suffer (ie. the cost of accommodation from the period when their tenancy is stated to start to the date that they actually move in).
Hopefully a stern letter the tenant will scare them in to moving out on time.
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