It is better to think about reported speech as reporting information somebody gave you, not as reporting words that someone told you. When you give somebody else - a new listener - that information, it must make sense to the listener now.
Consider the Original Poster's example. Lets us give a date that the original information was provided
We can deliver next week. (Said on the Monday 1st February)
He said they could deliver next week.
Now suppose it is still 1st February. You want to tell your friend what the company said. The potential delivery time is still next week. When you give the information to your friend you will still say next week, as in the example above.
Now suppose it is Monday 7th February and you want to tell a friend what the company said. The information that the company gave you was that the potential delivery time was in the week 7th-13th February. Today is the 7th of February. When you give the information to your friend, you will have to say this week, because this is the week of the potential delivery:
He said they could deliver this week.
Now suppose it is July, several months later. You want to tell a friend about the information you were given when you tried to buy the sofa in February (even though you did not buy the sofa). The information must be true, it must make sense to your friend now. The potential delivery date was 7th-13th February. You spoke to the company on 1st February. But we don't really care about the actual dates! We can help the listener understand which week by saying the following week. This just means the week after the time we are thinking about. The time we are thinking about is the time when the conversation happened:
He said they could deliver it the following week.
This is fine, because it still makes sense to your listener now.
If the information you were given makes sense to your listener now, your sentence will always be grammatical. It doesn't matter what words the first speaker used. It happens exactly the same way in your language too!
I hope this helps!