The answer is contained in your question and further implied although sometimes rather obliquely in some of the comments and answers.
The wooden de Havilland Mosquito used two Rolls Royce "Merlin" engines, with the exhaust stacks angled backwards because about the time of the development of that order of engine with that amount of power, it had been discovered that angling the exhaust stacks rearward gave a speed advantage in the low tens of miles per hour over the top speed of a 300mph-plus aircraft compared to when it did not have the stacks angled rearward. ie. there was something of a "jet" component already in the Mosquito with large quantites of hot air from powerful engines blasting backwards at full throttle.
EDITED: REFERENCE ADDED:
The below references about the "jet" effect I mention above, are not the two historical references I first discovered in 2010, which first led me to respond to the question. I have mislaid those references in house moves and computer changes. I have not read the below references but it is all I could find now, probably given that jet effects from piston engine exhausts are rather irrelevant, and doubtless gradually becoming of even less interest, in modern aviation whether in war or peace time.
Scroll down to "Ejector Exhausts". Note citations no.35 & 36.
cit.35: Price 1982. p. 51 (Bib: Price, Alfred. The Spitfire Story. London: Jane's Publishing Company Ltd., 1982. ISBN 0-86720-624-1).
cit.36: Tanner 1981, A.P.1565E, Vol.1, Section II (Bib:Tanner, John. The Spitfire V Manual (AP1565E reprint). London: Arms and Armour Press, 1981.
(end of the references edit)
Therefore I can't see much difference with pulling the Merlins out of a standard Mosquito and replacing them with gas turbines. And there is my answer to the first part of your question.
Obviously the presence of scorching hot exhaust gases from two piston engines providing that "extra" thrust didn't bother the wooden Mosquito.
Of course, one would need to consider exact mounting position for turbine engines on a Mosquito as they spew out a lot more, and hotter, gas than piston engines, but I don't see that as an insurmountable problem.
And how "hot" is hot? Was it the Canberra jet, a metal twin engined aircraft, which had the jet engines on the wings? I never read anything about, for example, the paint on any logos painted on the sides of that aircraft being scorched or melted off by jet exhaust. I suggest therefore, that even with the aviation understanding of the 1940s, that the Mosquito could stand as an example of a wooden jet aircraft that wasn't built, but could have been if the engines had been available.
And no, I am not suggesting the Mosquito would be capable of supersonic speed as a twin-jet. That issue has already been addressed in comments and answers.
VAMPIRE BRIEF EDIT:
The WWII-developed de Havilland Vampire was of course part-wood construction. It was used as the Royal New Zealand Air Force's fighter force for perhaps 20 years. The evidence of successful overall design was strong by dint of my personal observation of the aircraft screaming over my house for decades from the Ohakea AFB some 30 miles away. It's a moot point how much the wood contributed to the Vampire's success as a fighter, but it must contribute to part of the body of evidence that wood and jets can work together, even if only partially and at subsonic speeds.