First question is - do you have the logo without the text? If not, it can be removed using the resynthesizer + heal selection plugins. Look in the PCLOS repo
Then a way to add the text. I prefer a default text-along-path then path-to-selection and a fill with colour.
In this case you are probably better off with Ofnuts text along path plugin, then the same path-to-selection/fill selection.
5 down here: ofn-text-along-path.zip
https://sourceforge.net/projects/gimp-pa...es/scripts
unzip , comes with documentation.
I did this for someone else,
https://youtu.be/7fumw57Z2H8 first minute or so shows that plugin in use. Might help.
or this one for standard Gimp text around a circle:
https://youtu.be/pukUIefS0xI
Attached an example of what you might end up with, this using Ofnuts plugin. Might give you a clue.
edit: imgur decided to start working again, example looks like this:
https://i.imgur.com/jDF7pSR.jpg
There is a bit of green shading on the lettering, might be an arifact or you can do with a stroke path and offset it a tiny bit.
Edit#2: If you are making a new logo from scratch in that style, I think it will have come from Photoshop using Redfields Fractalius plugin. You can get a similar effect in Gimp using the gmic plugin and in the artistic section the Rodilius filter.