Well, you need to sort out what exactly the offer is. Usually employers don't want the employees hanging around through the notice period, so they dismiss them. Part of the overall payment needs to cover an amount that you are due for breach of contract (by the employer) in not giving you your full notice period. In that situation, your last day of employment (termination date) could be next Friday (say), and the P45 should follow pretty soon after that. It will state that your final day of employment was Friday 12th. That date is used by the Job Centre for registration purposes, and would enable you to a credit for the NI weeks if you are unemployed (though any due payment for Contribution-based JSA would typically be deferred until the end of your contratual notice period). There is, however, nothing to prevent you starting another job before the end of the period.
In more rare situations (typically for management), the employer wants the employee to stay at home and continues to pay them normally until the end of the notice period - and effectively prevents them from starting with another employer.
Ask your employer what your proposed termination date is.