Joint tenant wants to leave before end of term

One of my joint tenants wants to leave half way through the initial 6 month term. The other tenant wants to remain. What is the best way to proceed? If I agree that one tenant can end the tenancy early, their name will still remain on the tenancy. Presumably I will need to agree a surrender with both of them and then start a new tenancy?

Both tenants are liable for rent for the whole fixed term unless you allow them to end it early. If you do this, the all parties sign a surrender and the tenancy ends for everyone. You cant just release an individual tenant in a joint tenancy.

I would want to re-credit check the one who wants to remain and only agree if he can afford it. If so then you need a whole new tenancy with all documentation and dealing with the deposit. All this has a cost and you can charge the tenants, but only your actual costs.

Just out of interest would the procedure be the same if the joint tenants were outside of the fixed term and on a rolling periodic contract ?

No, with a statutory periodic tenancy, either tenant can serve notice unilaterally, but it does still end it for both. With a contractual periodic tenancy, it depends on the wording of the tenancy agreement.