How to solder a connector when spacing of PCB pads is not equal to spacing of component pads?
For a prototype, or to build 2 or 3 while you wait for the new PCBs to arrive, you can fix this, very carefully, with a scalpel.

Notice :


A group of (say) 4 pads will align well enough to solder. You might get away with 6 or 8 but I suggest 4 to tolerate some misalignment. Or 5 for 6 groups of 5 each.
You have a flexi-circuit connector.


Identify the 2 connectors in the middle, and run the scalpel between them, up to the / \ fanout section but stop before the really thin tracks from under the display. EDIT : Mark the groups with a pen before cutting!

Work outwards from there, making a set of 5-wide flexible connectors that will stand some misalignment.

As you get further out you will have to cut round the corners and most of the way along the angled sections, stopping before the root where the thin tracks are. 

Now you will be bending the outer segments outwards a bit, which will move them upwards; I hope the PCB fingers are long enough so that they reach. 

I would tack the outer end of each group in place before trying to solder them all properly.

If you DO cut a track you will have to scrape off some Kapton and resort to fine wire to repair it...

This is not guaranteed and may not work first time but I've seen good technicians make a splendid job of far worse...