Having an interline agreement is one thing - but you don't have an interline ticket so cannot avail of the "benefits" of an interline ticket. You're travelling on 2 separate tickets and checking through luggage is a service intended only for customers travelling on an interline ticket.

3 hours is rather tight - presumably the check-in for your Virgin flight closes 60 minutes before departure; you will have to make your way from Terminal 2 (where A3 arrives) to Terminal 3 (from where VS flights depart)....but it's only about a 10 minute walk so at least you don't need to deal with trains or buses. If there are any delays on your flight from Athens, or any delays or issues having your luggage delivered and/or any delays at the border, you're not leaving yourself much leeway to get to your Virgin flight.

I, personally, would never book a ticket like this. I do occasionally book separate tickets, but never with such a meagre margin between them. If you miss your VS flight, in all likelihood your ticket is entirely gone (unless perhaps it's an award ticket or a flexible - i.e. expensive - fare that allows you to make changes after being a no-show).

When booking separate tickets, you should be doing everything to MAXIMISE the time between the separate tickets - all the more so when the ticket you run the risk of missing could be very, very expensive to replace if you miss your flight.

(I usually arrive the night before my main, longhaul/expensive flight departs).

If you've already booked this, I would recommend travelling hand-baggage only.