I think that timber is technically felled, but unprepared wood. We have timber yards, where you can buy timber.
Wood is just the material, a tree is made of wood, the table is made of wood, the beam is made of wood. Maybe you are confused because we call a collection of trees a wood, or woodlands?
You can buy rough sawn timber, planed timber, etc, but it's all made of wood.
So your beam is probably a timber beam that is made of wood. It's both a wooden beam and a timber beam.
The reality is that most people will use both words somewhat interchangeably.
As your specifically talking about construction, in England we would almost always use timber, or timbers for the components.
- The roof timbers.
- Framing timbers.
- Timber framing.