If we're talking a cast iron radiator, it's a good process.
First you need to remove the old finish. You can get it sandblasted, or you can get in there with some sandpaper for a few hours. A wire wheel on an angle grinder can get out a good bit, or you might have luck with chemicals to strip them. If you do use chemicals, you'll have to wait for them to totally dry, then sand the bare metal of the radiator just to make 100% sure all of the residue is gone and the surface is rough enough.
Next, you'll want to clean with a deglosser, to make sure all the sanding residue is gone, along with any other junk in there. The paint won't bond so well without it, and with something that expands and contracts as much as a heater.. you want to make sure it's thoroughly clean.
Now it's time to protect your walls and floors. Get a drop cloth, cut a slit in it and slide it over the unit. Tape it to the walls, tape it to the floor, and tape up as much of the slit you made as you can.
You'll then need to prime it, and you'll have to chose the primer based on the paint you end up with.
Time for paint. Oil based paint should hold up, but you'll probably have better luck with spray enamel paint. Use many thin coats rather than few thick coats. Make sure to read the specs on whatever paint you decide on, make sure it can withstand up to 220 degrees - Just to be safe.
Alternatively, maybe you can look into building a radiator cover.
It is very unusual for Urethane to Buckle or alligator skin when applied over urethane. The reason for this type of failure is usually a contaminate on the surface. Grease, oil, wax etc. Did you use the same type of urethane from the same lot? After sanding, did you clean the dust off with a cloth that may have had some contaminates on it? I always use a clean cloth very slightly damp with water, or a tact cloth, never use mineral spirits. Give it a while to dry completely and put on the next coat. The only other problem could have been that the urethane was not mixed well. Always stir, never shake urethane. Don't mix water base urethane with oil base.
Best Answer
If you were to paint the surface it would cover and look more uniform than if you were to try and re-stain. Re-staining is more labor intensive in that unless you know the stain color it is hard to get a perfect match.
Without a picture to judge what is currently on the surface the best way to apply a paint coat is to start sanding. Hand or orbital (or both). Start with 80 grit and work through to 120. Be aware that this is most likely a veneered table top which may have only a thin layer of wood grain. If you sand through it you will expose a core or wood ply that may look different. Paint will cover this under layer but be aware. Use any paint type you prefer. An oil based paint will dry harder than other latex paints, but is harder to work with. Apply several coats.
If the stain color is even and no re-staining is required you could give it a light sanding with a 220 grit paper, wipe the dust off with a dampened rag of paint thinner, let dry and apply several coats of polyurethane (oil or water based, gloss, semi-, or matte finish).
Be aware that while you can varnish over paint it (in my opinion) will not look good. The varnish will give a light toned paint a red hue. Painting over a varnish is possible ,too. You must apply a de-glosser to the entire suface first. This allows the paint to adhere better to the surface.