Please help! Rescue goldfish--need to transfer into new tank?

You should move the fish to the new aquarium as soon as possible. The bowl won't be cycled any more than the new aquarium, but the aquarium at least will be cleaner and the larger body of water will be more stable and able to dilute the Goldfish's waste better, and presumably also be better since it has a filter.

Seeding the tank with filter media or gravel from an established tank can help speed things up, or you could try Tetra Safestart to do the same thing. Live plants can also really help speed up a cycle, but you should only add them if you want to leave them in long term. While Goldfish will eat most live plants, they should leave Java Fern and Anubias alone, and those can really help speed up the cycle. Then there are things you can do to make the cycle less dangerous while it's happening, such as 50% daily water changes, using Zeolite in the filter, and using Prime as your dechlorinator to detoxify Ammonia between water changes.

When moving fish from very bad water to very new, it's possible to shock them. So it would be a good idea to try to acclimate the Goldfish to the new water before adding them. In this situation, the easiest way would probably be drip acclimation. To do this, get the new aquarium all filled and set up ready for them to go in. Get a length of air line tubing and get a suction started (you can suck on the end to get this done), then tie a loose knot so the water is just dripping from it . Allow the dripping to go into the Goldfish's old tank, set below the new one, and allow the water to mix over an hour. Scoop out some water if you need to keep the old tank from overfilling. Once the Goldfish have been acclimated to the better water for that long, you can net them in place them in the new tank, leaving all the old water out so it doesn't pollute the new stuff. It might be a good idea to add a couple drops of Prime to the old tank before adding in the new water, too.

The goldfish will likely survive an in tank cycle. You can not speed up the cycl unless you have used filter media from a currentl cycled tank.

Add prime with every water change and make several partial water changes a week for 6-8 weeks, or until test show you that the tank has finished cycling.

Safely transfer the goldfishes to another place, clean the bowl put the fishes back in. Let the tank cycle. That should do it. :-)