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The really simple option is to loosen the nuts while the wheel is resting on the ground, and only raise it up on the jack once they are loose (not removed, just loose)

Otherwise, if you don't have brakes you will need to look at wedging the wheel or axle somehow. You can use a strap, or wooden wedges, or as @SolarMike suggested, if the wheel has holes in it, a pry-bar through the hole and wedged under the trailer will work.

But option 1 is really the easiest way - it works using gravity, there are no straps to slip, and you reduce your chance of injury. If the wheel still slips, add weight to the trailer or attach it to a vehicle with the brakes on, or ensure the trailer is on a non-slip surface (eg dry, textured concrete)

Yes, loosen nuts while the wheel is on the ground but first dowse them all with some PB-Blaster or similar penetrant oil like WD40 and let it soak in for an hour or so. That should make them easier to come off and not spin the wheel.

Do you have a long solid tiedown strap? Wrap the strap around the rolling tread of the tyre and hook it to a secure piece of the chassis.

If there's nowhere accessible, hook up a tow vehicle and lay the strap under its rear wheel.

This should prevent the wheel from rotating when you apply torque via a wheel brace or cheater-bar.

Once your tyre is replaced, make sure you clean the studs and nuts, and use grease or assembly compound on the threads to stop this happening again. It would be practical to clean the studs on the other side too before they seize. Remember the spare wheel's mount as well.

If its stuck that bad, it may be somewhat fused through corrosion. Over torquing is likely at some point to break the lug and nut off all together. When that happens it will be fast, and may throw you off balance.

I'd use a dremmel or similar device with a small diameter cut off wheel and put a slice on one or two sides of the nut, going along the long axis of the lug. Try not to cut into the lug at all, but enough so that you can break the nut of without damaging the lug.

Then it may be possible to get enough corrosion off the lug to apply a new nut.

If you are feeling cowboy and don't care about breaking the lugs off the wheel, you could try getting two lug wrenches and breaker bars like 1 inch steel pipe to give you greater leverage. Have your monkey hold one lug wrench torquing in the clockwise direction, while you do the other way on the offending nut. Non-negligible possibility for injury!

As a last method: Heat the nut up with a torch and should work to free the bolt. Replace studs while hot if you have to twist off the bolt because its that stuck. I'd replace the bolt and nut anyways.

For getting the torque on the bolt you'll have to sit it on a wood block and put a load on the trailer if it isn't heavy enough.

I use the wheel's momentum as an impact wrench.

I wirebrush it and hit it with Kroil, maybe a little heat and some more Kroil. Then score a line on stud and nut so you can see when it's starting to move.

If you are careless, this technique can injure you. I use a long breaker bar with big (1/2-3/4) socket size with an impact socket. I jack and thoroughly jackstand the wheel barely off the ground. I fit the breaker bar/socket on one nut, so the handle aims straight away from the wheel hub*. I also get safety glasses and a face shield.

I note the arc of travel the wheel/bar can travel without the handle hitting the ground. It should be no more than 270 degrees or so, or the handle is too short or the wheel too high. If the surface is soft or I care about it, I put a 2x12, wood crib, or old phonebook where the handle will hit on the right. Ready?

I rotate wheel/bar to the right briskly. WAP! The breaker bar hits the ground and stops instantly. The wheel has mass, and does not want to stop. It wouldn't if the nut were loose. So a great deal of force exerts upon the nut, socket and bar.

I am crazy, not stupid; so I do this exercise several times, starting at zero force to learn where to safely put my hands, body and face and keep the tool on. Increase force slowly until the witness mark starts to move. Expect tool breakage, though I've never broken a tool or failed to get a wheel off. I am careful and use top shelf tools. If you are careless, this technique can injure you.

Then I use Kroil to run the nut off and back on, to hold the wheel so if the last stud breaks, I don't get a wheel in my lap.

* Interesting science question, if I go the other way and have the handle cross over the hub, does that increase effective torque or decrease it? Probably a wash, but my way gives longer reach.