London is surprisingly green for a city of its size, and logging was allowed in places like Richmond (Great) Park right up to the turn of the century. In any event, I suspect Bob Cratchit would have mostly burned coal in his fireplace.
Samuel Pepys’s staff in the Navy Office were doing clerical work in the 17th c. I don’t have time to go through the diary but there are plenty of references to the working routine. Pepys eventually gave up writing the diary in the belief that he was going blind. Candle light made his eyes ache increasingly as he got older and the optical aids of the period could not cure his problems (which probably involved astigmatism).