One quick but critical point before getting into the details. Metal blades tend to be really thin, often as much as structurally possible, so this question really boils down to: can wood actually help you make an effective blade with even less metal?
I think the biggest issue is the use of a wooden core. Combining wood and metal is certainly nothing new (there are plenty of historical weapons that use metal sparingly) but this approach would undermine the advantages of both materials.
We can look at the fascinating (and very effective) wooden weapons developed by Maori communities before they had access to metal. Even thin wood weapons like the wahaika below that provide highly concentrated impact (seriously, these really hurt and can definitely be lethal) are still much thicker than sword blades.
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A wooden core would probably have to be quite thick to sufficiently handle weapon impact. A thin wooden core, or even just a thin wooden edge on a thicker wooden core, would raise a lot of durability issues when creating a sword-like weapon. Imagine applying metal film on thin wooden chips - I can't think of a way to make that simultaneously sharp and strong. You need to make it relatively thick.
What makes a good sword blade? The complete opposite. To cut effectively, you want very thin, flat blades (not a thick wedge). You would lose a lot of cutting capacity by making it thicker - consider wood axes, which are great at splitting wood but terrible at cutting. A combat axe will feature a much thinner flat blade like the Dane axe (11th century replica) below.
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In short: I can't imagine how this kind of design could be thin enough to cut effectively and while still being durable enough to be practical. You are much better off with a proper metal blade - even a small, thin one - mounted on a wooden body.
So, what do you do when you make an effective weapon with only a bit of metal? Throughout history and across the world, this is where pole arms really shine! As several others mentioned, the spear family is probably the thriftiest way to combine a bit of metal with a wooden body. There's a reason they remained the most common infantry weapon for millennia (vastly underrepresented in modern media, but don't let that fool you).
However, since they are perhaps the least sword-like pole arm, I wanted to add that you can reclaim some cutting capacity with larger (but still relatively small) metal blades. The glaive family is essentially various knives mounted on wooden poles - a good way to get something closer to a sword without much metal (and an opportunity to really optimize, if you need to).
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It's hard to pick a single best design without a full understanding of the context, since that will determine the right tradeoff between cutting capacity vs metal quantity, but something on the broad spectrum of pole arms would be a much more effective way to combine a bit of metal with wood to create a practical (and sometimes vaguely sword-like) weapon.
