Hello and thanks for using Just Answer.
My name is ***** ***** am happy to assist you with your enquiry.
I think you will appreciate your practical problem is a family one, rather than a legal one, but I can explain the latter to you.
You are correct in that under your Buy to Let Mortgage, you are forbidden to rent out to a relative. Nor are you permited to rent out unless the Tenant signs an Assured Shorthold tenancy. By letting out the property to your son, you are therefore breaching the Mortgage terms and if your Lender found out, they could issue possession proceedings for breach of the Mortgage terms.
Aside from the above, you risk giving your Son rights to remain in the property if no Tenancy is signed or rent paid. Without a written agreement between the parties, it leads to potential problems in the future, if for example, you need to evict your son and he is not willing to do so! With a written Agreement, at least both parties know where they stand and can avoid unneccessary arguments in the future.
As joint owner, you are of course entitled to gain access to the property, and indeed, you are permitted to use reasonable force to gain access if the keys are not available to you.
I hope this assists and sets out the legal position.