Anthropologists, for centuries, have gone into areas peopled by tribes who spoke only an unknown language, and became fairly fluent in a few years, enough so to write entire books on the linguistics of the people.
Although English and Japanese are unrelated, they share a lot of vocabulary. (English has borrowed a few words from Japanese, but Japanese has borrowed many from English.) You could say a lot just with that shared vocabulary.
The OP sounds like a plot point of the movie Caveman, starring Ringo Starr. Tribes who are just inventing language try to communicate:
*Atouk: [as Bork holds up the corpse of a slain bird-like creature] Ool?
Nook: [with modern accent] “Ool,” no. Food.
Lar: [looks to Atouk] “Fud”?
Atouk: [not knowing what Nook is talking about] Ool.
[others agree: “Ool.”]
Nook: [pointing to the campfire] That’s fire.
Atouk: [looking at Nook like he’s lost a few marbles; after all, Atouk just discovered and named it!] Haraka.
[Others repeat, “Haraka”. Nook puts his palm to his cheek, looking frustrated]
Atouk: [pointing to Nook] Bobo?
Nook: Friend.
Atouk: “Fend”? Bobo!
Lar: [slowly and exaggeratedly] Bobo.
[others repeat, “Bobo.”]
Atouk: [getting that Nook has different words, challenges him to a definition battle] “Alounda.”
Nook: “Love.”
Atouk: “Zug-zug.”
[Nook pauses, looking as if at either a loss for words or for MPAA rating. Lar and others begin to chuckle triumphantly]
Nook: [shrugging in defeat] All right… “Zug-zug.”*
Louis Gossett Jr. and Dennis Quaid figured it out: Enemy Mine Trailer - YouTube
Cite that JM’s example was the result of an entire group of people pulling together vs one translator.
Interestingly, one of the first Native Americans the Pilgrims encountered in the New World was Squanto, who spoke English.
I did this with the contractors for our house here in Taiwan. They didn’t speak English and my Chinese is nonexistant. I’ve done a bit of carpentry in the States so I knew the basics of what I wanted. My wife was translating at first, but it took longer to explain the concepts of what we were doing than just to draw picture and use gestures. As it was, we used a mixture of a few English words, some Chinese and some written, since I read Japanese.
We extablished the shared vocabulary by looking at something. They would say it in Chinese, I’d say it in English and then we would use whichever the other one would remember. Grammar was simplied into a pidgin English/Chinese level.
“Tomorrow. You, me come. 10 (point to watch.) OK?”
Steven Pinker discusses the process in Language Instinct were various groups of people are brought together. First they speak a pidgin language, then it developes into creole.
No. The loan words tend to be specialized and not what one would use for basic conversations.
Which he had learnt through being abducted and taken to England, where nobody spoke his language.
Children who move to a new country, and are sent to school there, rapidly learn the new language, without formal teaching. (Adults generally take a bit longer, even with formal teaching, bilingual dictionaries and the like, but they still manage.)
Me Tarzan.
It would appear that your parents managed not only cross-language, but cross-species communication, and even romance! 
This is absolutely right. I also point out Kenneth Pike (Kenneth Lee Pike - Wikipedia) who was famous for a demo he would give every year at the Modern Language Association meeting. He would sit at a table with a few props (a piece of fruit, a tool, a pencil,…) and they would bring in a native speaker of a language Pike had never learned (there always were some) and he would start out by learning the words for the props by pointing. He would then try to converse with the native speaker. Within a half hour he would have picked up enough grammar and vocabulary to have a simple conversation.
Hardly anyone has that ability, but we all have some ability to learn a new language. It would take a lot longer than a half hour, but certainly it would happen in under a half year and we most likely would develop a non-grammatical amalgam of our two languages which, as noted, is called a pidgin. Our children would mold it into a creole that has a full grammatical structure. Because kids are geniuses at grammar, a facility that nearly all of us grownups lack.
Incidentally, New Guinean “pidgin” is actually a creole. While Haitian “creole” is mainly just a dialect of French.
Specialised words like “Baibai” and “Sankyu” (from English “bye bye” and “thank you”)?
I fail to see how it would be a problem. Even if one of the two persons is incredibly bad at languages or really lazy, the other could simply learn the language of the first. People don’t need book or a formal education to learn a language. In fact, bokks and a formal education are used because the more efficient direct communication isn’t available.
I bet that within five years, cell phone technology will render this whole question moot.
Not if they were on a desert island with no electricity. The cellphones would quickly run out of power, and probably wouldn’t have a signal anyway. Satellite phones would have a signal but would still need power.
I’m completely amazed that this is even a question. Of course they would! Good grief, learning is a language isn’t that hard!
Anecdote: When I was doing my TEFL certificate they gave us a lesson in Polish just to make us understand what it feels like to be a learner. They did the entire lesson through Polish, not a word of English. By the end of the hour we had learned IIRC some genders, the words for “I”, “you” and one or two forms of the verb “to be”.
In second grade my fresh-out-of-college 22 year old teacher was faced with a challenge. A new student enrolled at our elementary school, a child newly arrived from Taiwan who spoke no English. None.
The teacher picked me and another student to partner up with the new arrival. The three of us were huddled off into the cloak room with instructions to focus on teaching/learning the alphabet. That was done by the end of day one, to reasonable reliability.
We spent the rest of the week pointing at things and exchanging words in the languages. Classroom items. Clothing. Body parts. Shapes. Colors. Whatever we could think of. By the end of that week the teacher had sorted out whatever additional ESL instruction was needed and my teaching partner and I returned to the fascinating world of learning subtraction with the rest of the class.
Our student was a bright kid and learned English rather quickly. He was communicating in short phrases in a few weeks. Complete and grammatically correct sentences in 3 months or so. By the end of the school year you really couldn’t tell that English wasn’t his native language. Well, I as a kid couldn’t tell.
I, on the other hand, didn’t remember a single word from my cursory Chinese lessons.
Note the word “tend” in my sentence. That you can find 20 to 40 common words (which “sankyu” is not) is simply nitpicking.
You originally stated that English and Japanese share “a lot of words” in common because of the loan worlds.
That’s simply uninformed bullshit and shouldn’t be on a site fighting ignorance.
Spanish, French or German share “a lot of words” in common with English. Japanese does not.
The number of English loan words in Japanese is such an infinitely small part of a person’s working vocabulary that their existence would make absolutely no difference in this scenario.
Of far, far greater importance would be the words they would have remembered from their eight plus years of study of English. For the purpose of the hypothetical, we are assuming they don’t remember it at all, so knowing a half a dozen or two words ain’t going to help.
The technical term for this process: “Pidgin”.
A “pidgin” language becomes a “creole” when it stops being a mere method of two people communicating without a language in common, and starts being a "native language (say, if these two have kids who grow up only hearing “pidgin” - they very soon will invent al sorts of shades of meaning, making that simplified “pidgin” into a full-fledged “creole”).
Can they make a rudimentary lathe?