This is the bit you thought would be nearly impossible, but it's not actually very hard.

With the front plate flat on the table, first replace the two minute-to-hour wheels,

Referring to the photos you took earlier, insert all the arbors into their pivot holes in the reverse order to that in which you removed them.

To make sure everything is properly in place, check that every wheel except the contrate wheel engages with a pinion. (The astute reader will have noticed that in the photos, I had the contrate wheel and its predecessor the wrong way up. But it gave me more practice in aligning pivots with pivot holes!)

The square main spring winder and minute hand adjuster are taller than any of the pivots. Lower the back plate onto these and onto the four posts joining the front and back plates.

At this stage, none of the pivots will be lined up with its pivot holes, but with the back plate at a very slight angle, it will be resting on one of the pivots. Gently manoeuvre this into its pivot hole.

Repeat for each pivot in turn. Once all the pivots are in their pivot holes (double check that they indeed are) you should be able to press the back plate down onto the four pillars as far as it will go.

Holding the back plate firmly in place try gently spinning one of the wheels and check that all the others turn completely freely. You can now reinsert the locking pins.