I'll talk through the ideas as they occur in my head if it sounds baby box or insulting better that than miss out a crucial step. Miss out all the steps and answer the last question first. Are the offside lights on the car now working?
On a car the term earth is interchangeable with negative (the return current is through the body) On a trailer especially a folding camper or caravan this should not be so. So the terms negative return and earth may all get a bit jumbled up.
So what has happened occurred after you riveted the light back on.
My first thought was that pop riveting on the number plate light it had splayed the contacts to the bulb (Presuming it was a festoon bulb) that would stop the bulb from illuminating.
As the front off side light is now not working that theorem hit the bin.
I'm now expecting the live to have shorted out on the trailer body. Full marks to those saying that the negative shouldn't be earthed to the Dandy body but should and is are two separate words.
Drill out the pop rivets and separate the light fitting. Does the fitting now light?
If yes congratulations I'd guess that the live earthed on the the body. Possibly the body is running as negative(earth), possibly the earth on the other side of the fitting grounded on the body as well so the current went through the body (of the Dandy) rather than the bulb.
So in this case tidy up the wiring on the fitting and re rivet.
You have removed the fitting and both lights are still out.
I'm hoping most of the above is still true but you have blown a fuse.
At this point we start to need a multimeter. Do you do multimeters? Even if you don't this is easy the multimeter will cost under £10 and it's a basic step by step test sequence. Also you will have gained a little confidence and still have the multimeter. If an auto electrician costs you less than £50 just to say hello you are doing well.
I'm presuming 12N electrics [You must be registered and logged in to see this image.]
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So you need to check if you have a supply from the 12N fitting. You need to check from pin 5 to pin 3.
Selecting DC voltage on the multimeter we want a reading of 12v between the pins. Putting the +ve probe (red) to pin 5 and the -ve (Black) to pin 3.
No voltage 1st did you switch the lights on? 2nd practice with the fog light supply (terminal 2) to check you have the right technique for measuring.
So still no voltage. Blown fuse is the favourite. Is there a separate fuse after the take off for the 12N socket. If not main fuses. Are the offside lights working on the car?
See how you get on with this. Any questions ask.