after WW2 there was a shortage of many everyday items in the UK, and rationing continued for some years after the war (sweets didn't come off ration until 1953, for example). There was a general attitude of "make do and mend", getting by without luxuries, "putting up" with things - even new items were often poor quality and simple (post-war furniture for example was called "Utility " furniture). Thus, whenever you wanted something repairing, such as a household appliance, a bike, a car etc - and not many people had cars! - you would often be told, "Sorry, I can't mend it - you can't get the parts, you know".

So this phrase a) echoes what people would have been used to hearing every day and b) makes the situation more ridiculous by taking a material which you would think would be plentiful (i.e. it's not "you can't get the steel/cement/spare parts" etc, but wood - and it also fits well into the context of the episode. When most people had coal fires you often used to hear people joke, "Don't put so much wood on the fire - it doesn't grow on trees, you know!" - a jokey play on the phrase "money doesn't grow on trees", i.e. you have to work for it, not just pick it up).
[When my children were young, they used to love an American children's book called "Treehorn's Treasure", in which money actually does grow on trees!).

Even today, you often hear people say "You can't get the staff, you know", as a jokey reference to somebody's inefficiency - could be your spouse, neighbour, workmate etc - in a "folk memory" of this type of phrase. THe Goons were fond of imitating "jobsworths:, petty officials and unhelpful tradesmen, so this phrase suited the tenor of the show.

HTH!