So, in the US, we make ample use of the word "just" in a context such as: "I just finished my homework." (I finished my homework very recently -- perhaps immediately preceding this statement) or "He had just gotten his driver's license when his father's car was stolen." (pointing out that there was very little [if any] time between the two events) I only recently learned that this is, apparently, an Americanism. Is there a better, more universally accepted way of expressing this? I feel that "recently" is not quite "recent" enough.
For the past perfect tense you have a few options:
scarcely: "They had scarcely made this resolve when a feeble cry arose from a dark object that floated rapidly by." -- Edgar Allan Poe, The Narrative of Arthur Gordon Pym.
barely: "We had barely completed this work when the Commander, the Captain, Marvin, Borup, and Esquimos came in." -- Matthew A. Henson, Matthew A. Henson's Historic Arctic Journey.
hardly: "...but he had hardly reached home when Frank, who had been sent after him, delivered this note..." -- James Boswell, The Life of Samuel Johnson.
The only Americanism I see in the sentences you cite is the use of just with the Past Simple. In British English, just is usually used with the Present Perfect tense, which is considered the correct thing to do according to grammar books.
As for another word to replace just, I can't think of one, just describes exactly what you want to express.
"We had sat down [only] moments before the remaining guests arrived".
"He had been shot that very minute".
"Seconds before, we had witnessed the strangest scene unfolding before us".
"A few moments ago, our lives were changed forever".
"We only got in three minutes ago".