The nearest study measured preference, not a boost in thinking
The one study that actually involves wood looked at side-wall colours in multi-screen personal learning spaces used by students [1]. It asked which colours people preferred and also tracked visual fatigue, attention consumption and working memory, with media multitasking treated as a factor. The headline result was that wood side walls were judged the most suitable environmental colour [1].
That matters because "most suitable" is a judgement about how a space feels or which finish people favour. It is not the same as showing that wood made anyone concentrate better or hold more information in mind. The study does connect wood, attention and working memory in the same setting, but its reported conclusion is about colour suitability, so it cannot carry the stronger claim that wooden elements sharpen attention during multitasking [1].
Multitasking itself is what strains attention
The other studies help explain why the question is worth asking, even though they say nothing about wood. One line of work finds that heavier media multitasking tends to divide students' attention, while people who process information differently showed higher working memory capacity [2]. In other words, juggling screens and streams of information is the condition that puts attention and working memory under pressure.
A separate review of athletes found a similar pattern under dual-task conditions, where people perform a motor task and a cognitive task at once. Even athletes familiar with time-constrained, multitasking situations showed a cost in motor and cognitive performance [3]. The takeaway is that doing two things at once reliably taxes attention and working memory, which is exactly the situation in which a calming or restorative environment might, in principle, matter.
What would actually answer the question
To show that wooden elements improve attention and working memory during multitasking, a study would need to compare a wood-finished virtual workspace against a non-wood one while people perform demanding concurrent tasks, and measure attention and working memory in both. None of these studies does that. The wood study sits in a learning-space context and reports a colour preference [1], while the multitasking studies examine attention and working memory without any wooden or virtual-environment element [2][3].
So the practical reading is cautious: wood may be a preferred finish in screen-heavy study spaces [1], and multitasking is known to strain attention and working memory [2][3], but the bridge between them, a measured cognitive benefit from wood during multitasking, has not been demonstrated here. Treat claims that wooden virtual elements boost focus as plausible but unproven.
Sources used in this answer
The study of preferred side wall colours in multi-screen personal learning spaces based on students' visual fatigue
In multi-screen personal learning spaces, students rated wood side walls as the most suitable environmental colour, with the study also examining visual fatigue, attention consumption and working memory alongside media multitasking.
Attention and distraction online
Heavier media multitasking tends to divide students' attention, while individuals who processed information differently showed higher working memory capacity.
The acute and chronic effects of dual-task on the motor and cognitive performances in athletes: a systematic review
In a review of athletes, dual-task conditions produced a cost in motor and cognitive performance even among people familiar with time-constrained multitasking.