A Pantone is a Pantone when it's a "spot" color.
Make sure your Pantone swatches have the color type "spot color". Even if you color swatch says Pantone XYZ, it doesn't make it a Pantone; if it's not a spot color, it will be considered as a process color (CMYK).
Conversion during export
If you are 100% certain that your swatches are "spot" then maybe the issue is how you prepare your files and what profile you're working with.
Your color mode should be set to CMYK.
When you export, the section "output" has a setting for color conversion. Verify that this is set to "no conversion" when you export a PDF otherwise your file might end up in CMYK or RGB or Grayscale.
How to verify a PDF
If you export your files as PDF, be careful to not convert them to CMYK or RGB inthe process.
One way to verify your file (if it' a PDF), is to open them in Adobe Acrobat Pro and find the Output Preview in the tool sets.
Then uncheck all the Cyan-Magenta-Yellow-Black and see what's left on your layout. Logically if you used a spot color Pantone, that color should also be in the list of inks. If you don't see it then it probably means 1) you didn't set your swatches to be "spot" or 2) you exported your file and converted it in the process.
If you sent an .ai or .eps
If you sent the original Illustrator file,there's no reason why your colors appear in CMYK if you verified the details mentioned above. The issue is somewhere else.
If you sent any raster file (png, jpg) then it's possible there's no spot Pantone left in your layout.
Your images being neon
That does sound like some RGB issue. When you work on a print project, forget about RGB unless you know for sure the printer can handle it.
Print projects are usually in grayscale, Pantones or CMYK. The color mode you use is very important.
The proof you got also depends on the machine that printed it. If you got a digital proof, it's certainly a possibility it's not 100% calibrated and the colors are often more "luminous". Since you used an uncoated version of your Pantone preview, the colors could look different than what you expected.
Your previous printer
Printers often fix designers' files without telling them because they don't always have time to wait for the revised files because of the printing schedule or don't have time to educate designers on this! It's possible your previous printer simply fixed your file without telling you. If all the others say the file is in process, it might be a possibility. Maybe you simply changed the "output" color conversion while doing your export in PDF without noticing it on the new files, these things happen!