This answer is mostly Stiv's work, with the final answer filled in by me. From Stiv's answer, we know:

The word they are thinking of is RICH, as in "rich tea biscuit", "filthy rich", and when mixed with "Boe", "brioche". The letters from TEA, FILTHY, and BOE form a pool of letters ABEEFHILOTTY.

"You know, if you take the letters from all three words, you can also spell my chosen name and your nickname" pointing at the one on his left. "with only three letters left over. Now take those left over letters from your place of birth, where I once worked, and you could say you were blue!"

The speaker here is OLAF Stapledon, born William Olaf Stapledon but his chosen name was Olaf. He is a writer who worked in Belgium from 1915-1919 as an ambulance driver. The one on his left is HETTY Verolme, nicknamed Hetty, born in Belgium. The 3 leftover letters are BEI, and when removed from "Belgium" give GLUM, a synonym for "blue".

The other two look blankly at each other before one says:

"That's interesting," says the one who hadn't spoke yet, "I was born in the same country, but those left over letters spell my name."

The last speaker is Silvy De BIE, also born in Belgium. She is a musician, which fits with the hint and the tags.

"What a perfect storm of coincidences!" says the initial speaker.

The one who was last to talk smiles at that.

Silvy De Bie smiles at "storm" possibly because her song "Angel on my Shoulder" has the lyrics "You're an angel on my shoulder, My shelter from the storm".

Who are the three, what is the word being guessed, and which is the time traveller?

The three are Olaf Stapledon, Hetty Verolme, and Silvy De Bie. The word being guessed is "rich". The time traveller is Olaf, since he lived from 1886-1950, so he would need to have time travelled to the future to meet Silvy De Bie, who was born in 1981. If instead Silvy time travelled into the past to when the other two were both alive, then the initial speaker (who was not Silvy) would not have known they were all rich, because they would not have known Silvy.

This answer is mostly Stiv's work, with the final answer filled in by me. From Stiv's answer, we know:

The word they are thinking of is RICH, as in "rich tea biscuit", "filthy rich", and when mixed with "Boe", "brioche". The letters from TEA, FILTHY, and BOE form a pool of letters ABEEFHILOTTY.

"You know, if you take the letters from all three words, you can also spell my chosen name and your nickname" pointing at the one on his left. "with only three letters left over. Now take those left over letters from your place of birth, where I once worked, and you could say you were blue!"

The speaker here is OLAF Stapledon, born William Olaf Stapledon but his chosen name was Olaf. He is a writer who worked in Belgium from 1915-1919 as an ambulance driver. The one on his left is HETTY Verolme, nicknamed Hetty, born in Belgium. The 3 leftover letters are BEI, and when removed from "Belgium" give GLUM, a synonym for "blue".

The other two look blankly at each other before one says:

"That's interesting," says the one who hadn't spoke yet, "I was born in the same country, but those left over letters spell my name."

The last speaker is IBE Wuyts, also born in Belgium. He is a musician, which fits with the hint and the tags.

"What a perfect storm of coincidences!" says the initial speaker.

The one who was last to talk smiles at that.

Ibe Wuyts smiles at "perfect storm" because of his song "Perfect Storm".

Who are the three, what is the word being guessed, and which is the time traveller?

The three are Olaf Stapledon, Hetty Verolme, and Ibe Wuyts. The word being guessed is "rich". The time traveller is Olaf, since he lived from 1886-1950, so he would need to have time travelled to the future to meet Ibe Wuyts, who was born in 2002. If instead Ibe time travelled into the past to when the other two were both alive, then the initial speaker (who was not Ibe) would not have known they were all rich, because they would not have known Ibe.