Assuming that the block size fork goes through, will the Bitcoin developers use the alert key to tell people to upgrade? The block size increase proposal currently under discussion would require all clients that process transactions to upgrade.

Otherwise, pre 0.11 clients would reject these larger blocks, leading to a possible double-spend attack against these merchants after Mar 1, 2016.

Will the Bitcoin developers use the alert system to warn these nodes to upgrade?
Yes, according to Gavin Andresen via Reddit.

Even if they didn't to do so, Bitcoin Core already has two other mechanisms which will produce automated alerts as the hard fork approaches: one which will take effect before the hard fork occurs, and one after (as per BIP-50).