I see, well it looks like you have what's called a euro cylinder lock, which is what the thumbturn is part of. You need to take that euro cylinder out, then re-fit your new door handles and their back plates, then slide the euro cylinder back in (from the inside).
This video shows you how you do that:
The kind of handles you need are something like this I think:
• Backplate length 175mm, width 44mm • Handle length 122mm, projection 68mm • Supplied with fixings
www.toolstation.com
But as that youtube video says there are various different sizings according to how thick the door is and the distance between the hole for the lock cylinder and the hole for the handles. So it would be safest to take your old handles in to a shop and make sure you get new ones that are the same size. They would be able to help you make sure you do that.
What I don't quite understand is how you got the old handles (with back plates) off, without removing the euro cyclinder+thumbturn. Did they have an unusually large hole that is bigger than the thumbturn?