Does a SQL Server update nvarchar statement overwrite the same address on disk if the new value is the same size?
There are so many aspects to this, but the short answer is that you have to expect that old values can be on disk (as this is from what I understand what the question boils down to).
So, this falls under the same category as below question, with the same answer:
Q: Can we protect from a Windows admin to be able to do anything inside a SQL Server and see all data in a SQL Server?
A: No. But you can try to make sure that what they read appear as garbage to them. I.e., a suitable encryption implementation.
If you are truly interested in this, I suggest you first read up on the storage architecture of SQL Server. Without that, you won't be able to make heads of tails of the answers. I.e., make sure you understand database files, pages, transaction logging, heaps, B-trees etc.
The update can be done in place, on the same page. That goes for an index as well. If you modify the key of an index column then the row has to move (DELETE/INSERT internally). If the row doesn't fit, then you get a forwarded record or a page split. Another factor is that after a delete, (and an update can be delete/insert, remember), the old row will initially still be there as a ghost row. But even after that, the page isn't &quot;compacted on delete&quot;, i.e., the storage reclaim is done when the &quot;free&quot; space is needed at some later time.
And then you have transaction logging.
And backups.
If you have versioning enabled, then old row versions are in tempdb and cleared up when not needed, after some time.
And the added complexity how other, perhaps less basic, technologies affects the picture. I'm thinking of things such as:

Columnstore indexes
Memory optimized tables
Change Data Capture
Replication
HA solutions such as Availability Groups
And whatever more I could come up with given some more time to think about this

I think you get the picture by now. No, you can't claim that the old data is gone. Encryption might be an important component to this, depending on your particular situation.