Do I have to return my BT home router? My broadband contract started on the 10th of October 2019.
Furthermore, how can I stop the services after the 12 months contract that I have? My contract ends on the 10th of October 2020. Do I have to do something, or it will stop automatically?
No, you don't have to return your hub. Unless you call and cancel your contract, you will just move to a rolling contract with no discount. If you intend changing provider, do not cancel your service, the new provider will do that.
If you received your BT hub prior to 13th Dec. 2019 you do not need to return it.
If you are moving to another broadband provider at the end of your contract you just need to take out a new contrat with your chosen provider and they will inform BT and cancel your BT Broadband contract.
If you are not going with another provider you will need to give BT 30 days notice to cease your line.
If you do neither of the above your broadband will continue on a 30 day rolling contract. When this happens any discounts that you may have had for being in a minimum term contract will cease and you will pay the full price for the broadband package you are on.
Following my question, I would like to mention that I will change flat before the contract ends. Should I apply to change the address and then to end the contract either by change provider, or either by just end it?
If you are still in contract you will be charged cancellation fees if you move provider before the contract has ended. The fee is usually what is left to pay till the end of contract. Some providers may pickup that bill to get you as a customer but you will need to check.
If you do a "home move" that will result in a new phone line contract starting. Your broadband contract end date will stay the same as long as it has more than three months to run if not a new three month contract will apply.
You will need to work out what is best for you.
EDIT: See link
Furthermore, how can I stop the services after the 12 months contract that I have?
I'd check your order confirmation, BT have not done 12 month terms for a while now, back in October 2019 the standard term was 18 months.