If the landlord is planning to repossess the property for the refurnishing work, then they will either need to legally evict you or come to a written agreement allowing you to move out and then back into the property once the work is complete.
Yes, landlords must give a minimum of two months’ notice to terminate the tenancy.
If you do not vacate the tenancy at the end of the notice period, the the landlord will be able to apply to court for a repossession order. Once the landlord has this, they can get a warrant for bailiffs to repossess the property. This whole process takes several months on average, so if the landlord served a valid notice expiring 20th December, and you just stayed at the property until Jan or Feb, it’s quite likely the landlord would be unable to do anything about it.
This is not a good idea though, because means your current landlord would not give you a good reference when you moved. And in general it is always best to come to a consensus position on move-outs like this. If they are truly just refurbishing, then they will probably want tenants when it is over, and if you have been good tenants, then there’s no reason for them not to let the property to you after the work is complete.
It sounds like the landlord has not yet served valid notice, however, and 2 months’ minimum must be given. Therefore, it sounds unlikely you’ll have to leave before Jan/Feb.