It's unlikely a data protection request will take months, but I don't have any first-hand experience to speak from.
Let me clarify the process. The Notice of Enforcement, issued for a traffic debt, expires after 12 months from the date of issue. However, this expiration only terminates the enforcement, not the liability for the debt.
If the PTO puts a 21-day hold, then that is at their discretion; they can arbitrarily revoke the hold at any time.
I am not hopeful a PE2 will succeed, and if it fails, then that will re-ignite the enforcement, and that's £514. My understanding, and someone else might put me right, is that PE2 only suspends enforcement, so the suspension is lifted. If the PE2 fails, the process continues from where it left off.
If you leave things where they are, then you might get a few more text messages, but enforcement will be difficult without your address. I see your previous address is a flat, currently empty, so getting anything there is unlikely to come to more than text messages.