The conditions you've given are valid for making languages more compatible
The historical example here might be comparing say, Latin, to any given Latin-derived language, such as Spanish, for instance. Now, the grammatical structure of the languages aren't the same at all and many of the words have been swapped around or changed, but if a Latin speaker attempted to make a point to a Spanish speaker, there'd be a fair chance that even without any knowledge of the other language, rudimentary communication would be possible. And, since they're fundamentally the same, a pidgin would be very easy to form, assuming that they didn't feel the need to just pick up the other person's language.
What would make things more difficult? Things like:
- Time gap - the longer the time the languages were separated, the more pronounced the difference will be. Dial back around 1500 years, and the predecessor to the current English is nearly incomprehensible to modern day speaker.
- Languages with no shared ancestors - Take for instance Spanish and German, which are derived from different ancestral languages. It's a lot easier for someone with the knowledge of French, a Latin derivative, to learn Spanish, another Latin derivative
- Languages with odd vowels - Certain languages (Navajo, for instance) have very specific sounds in their language that is almost impossible for someone who did not grow up speaking the language to sense the nuance of, much less pronounce
- Grammar structure - Here's another Latin example. Because of the way Latin works, you can technically put a Latin sentence in any order and it'll mean the same thing. The fact that words means different things based solely on their place in a sentence is a novel concept to certain languages.