I am summarizing the entire process of replacing an UG filter stepwise. Please correct me if i am wrong anywhere.

Step 1. Unplug the heater and let the tank sit for an hour

Step 2. Unplug all the filters and pumps

Step 3. Remove 50 percent water into a bucket1

Step 4. Remove the fishes and keep them in that bucket1 of removed water

Step 5. Add an airwood connected with an air pump into the bucket of fishes

Step 6. Remove the remaining 50 percent water into another bucket2

Step 7. Remove the entire gravel,decorations and filters and put them into the bucket2

Step 8. Remove the old under-gravel filter

Step 9. Set up the new under-gravel filter

Step 10. Put the gravel back from bucket2 into the tank and lay it evenly over the plates of the new under-gravel filter

Step 11. Add 50 percent fresh tap water after treating with anti-chlorine into the tank

Step 12. Add the filters and decorations into the tank

Step 13. Add the fishes into the tank from bucket1

Step 14. Add the remaining 50 percent water from the bucket1 where the fishes were kept

Step 15. Wait for 15-30 minutes to settle the water

Step 16. Start the filters,pump and heater

RE Step 1
Why are you waiting an hour after the heater is turned off? Heater doesnt take that long to cool. You will get a large temp fluctuation and it can shock your fish.

Turn everything off, drain water into bucket 1, catch fish and transfer to bucket 1, add an airstone to keep them comfortable, i would also be inclined to add the heater if you think the job is going to take you a while. Remeber though you will need to let the heater have a 15min cooldown before you move it back to the tank (cool to touch).

Step11
Correct temp new water will be better to add also, your heater will not have to initially work as hard, nor will it stress your fish too much

Step 12
You will need to make sure the bucket 1 temp is similar to the tanks temp or you could shock/stress your fish.
Turn on all your tank equipment here, do not wait for the last thing, water doesnt need to settle, the sooner you restart everything, especially the filter, the better.

You NEED a test kit to test the water for the next few weeks. Major upheaval in your tank could cause a cycle issue