Hi. I am not sure if this is helpful , but I have used these “places” for wood in the past. I’m not in the same country as you so I don’t know how much is transferable to central London…. But if it’s not a very long piece you can ask a construction store or wood supplier to buy cut offs. In many places you can buy a whole beam and people go and get their specified size and then the rest is tossed ( I have seen containers and asked to buy the “wood” trash) . You could also look for it at a renovation station. Again this might be different from country to country, but there is often a place for reusable materials in renovation stations / recycling senters. Gumtree or a used goods store. I also think there is a huge difference between supply stores, some have cheaper materials , but you might have to stain it yourself.