You can move the fish and cycled filter to the 20g from your 5g and run the old and new filter parallel for a month and let the old filter's cycle run the 20g while it establishes the cycle in the new filter, after a month the old filter can then be put back in the 5g. And filled and both are good to go. This gets you the additional space immediately although you can't really keep both tanks at the same time immediately.
Safest way I know to get both filters cycled and up and running without risks to fish.
I use some
sponge filters. I have a 10g for
quarantine and an extra sponge running on another tank, when I don't need the10g It's drained cleaned and dried, and the filter moved to another tank to keep its cycle.
When I need the quarantine, I fill the 10g,
dechlorinate, then move the filter over again and it's good to go immediately.
If you have to have both running at the same time and don't want either shut down, you'll need
ammonia for the new tank, moving substrate or sponge squeezing will only get you so far, maybe a small percentage of the colony then they need food (ammonia and
nitrite) to multiply. That way functions like a bacteria in a bottle. It will give you some of the colony as a starter, but not enough to run the tank with fish it will speed cycle though. That's called "seeding".
What I would do, move the sponge from the 5g to the 20g and the fish, and let the cycle colonize over to the new sponge for a month or so while the old sponge does the work, then move it back and you'll have both running and can move fish around or buy new fish to stock either.
The 5g can be completely cleaned and dried out for that time, it doesn't matter. The cycle is in the sponge filter and it's just giving it time for the bacteria colony to split and colonize from the old sponge to the new sponge.
This way you get to use the larger tank right away. If the filter can handle the fish load on the 5 gallon, it will handle the same fish load on a 20 gallon also.
You can also move the filter back after 15 days and test the 20g to see what it's doing and if its done. the new filter should cycle quickly since it's starting with all of the bacteria from the 5g tanks filter, millions of bacteria all multiplying, it's not like your first cycle where 1 becomes 2 then 2 becomes 4 and takes forever to get to 1 million becomes 2 million ect. The colony will size up quick starting with a big colony and take over the new sponge quickly.
And if you think you may want to go to a 3rd tank at some Point, run an extra sponge filter on the 20g, then you'll have one ready to move on day 1 if you do get a 3rd tank. Nothing wrong with more filtration anyways.