If someone could claim (and they did) to be pen-makers to the Queen then they did provide the Queen with her pens for a period of time. They may not have been the sole providers, but would have been a real provider of pens.
As to the holder, I rather doubt she used a prosaic wooden holder. Anything's possible, but if you consider any woman of any means had a writing set with a holder usually of silver or pearl or some other special material, then I doubt the Empress of India was using a common, wooden, holder.
I would not be surprised to learn that she may also have used gold nibs as well as steel. I'm sure she had writing sets scattered all over her residences.
It's obvious I have no specific information, just an idea of what people wrote with at the time. Have you tried the Birmingham Pen Museum? They may have more detailed information.
“When the historians of education do equal and exact justice to all who have contributed toward educational progress, they will devote several pages to those revolutionists who invented steel pens and blackboards.” V.T. Thayer, 1928
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"No one is exempt from talking nonsense; the mistake is to do it solemnly."