I have done a few playfield swaps, and also dealt with one identical to your with the looms cut.
For a normal playfield swap (it all from an old field to a new field), the method mentioned using a piece of cardboard (or ply) somewhat bigger than the whole playfield is what I used. Assuming you've dealt with the top side (generally pretty easy on a pre-DMD game), you start on your old one by taking one hundred and twelve zillion photo's of the bottom. Then a couple more to be sure. And later when you still can't work something out, you PM a member with the same game!
next, you begin removing every screw for each component, leaving them basically in place in the process. Some components might be large and cumbersome - for those, remove them after tagging the wires so re-installation is easier.
Eventually you will have an old playfield with all the bottom components sitting in place, but able to be moved. The next part is easier with a helper, but not too hard on your own. You slide your cardboard slowly from one side between the components and the old playfield. Bit by bit it slides along - and then you'll start finding screws you missed, so remove them. Soon the cardboard will be between the old field and the components. Now slide that lot over on top of the back of the new field, and slowing slide the cardboard out. Now you have the components sitting pretty close to their correct spots, and in the process did not mangle and twist wiring which can easily happen.
Make sure you pay attention to screw sizes though - some components have thicker tabs (ie say a sling hinge) than others (ie lamps). The difference might not be much, but a screw for a think legged component used in a thin one could go through - the ultimate disaster.
As for the one I had with the looms cut - I sate with the old and new side by side, and tagged with a number every wire and it's mating component on the old then the new, and then cut them off and swapped the loom from the other field over. Took a long time to tag everything, but as the components on the one with the cut loom were much beetter, it was a better solution than doing the playfield sawp method and then swapping components back.
EDIT - I must have been typing while Wotto posted - see his suggestion!
