My understanding is that a pidgin is a language using the simplified grammar of one language with the vocabulary of another. South Seas pidgin uses English words within a Polynesian type of structure.
You get Gaelic/ English phrases in a similar way sometimes. "My head's at me" as a way of saying "I have a headache"
(I'm happy to be corrected on Celtic languages- I'm no expert)
The point I want to make is that English itself is a pidgin trading language. Everyday English uses minimal grammar compared with formal German or Latinate languages. All the case and gender agreement has gone, word order hardly matters, and we use vocabularies from anywhere.
"The knight came in through the door" is all Anglo Saxon.
"The cavalier entered via the portal" is pure Latin
English developed in the Midlands, on the borders where Anglo Saxon Speakers met Brithonic languages, Danish, Norman French, and Latin speaking lawyers and clerics.
The dialect of English that developed would depend on the ingredients in the local cocktail.
It's interesting that so many differences should have survived, and that most of us are to some extent bi-lingual. I write this in Oxford (the standard) English.
In speech you would hear a more Northern grammar and idiom despite the fact that I have never had a Yorkshire accent.
I speak North Midlands English. Scots speak Scottish English.
Cheers
Dave