I spotted a 'missing' bit of chimney in the house I bought.
Different to yours, the chimney was present on ground floor, one half had been taken out on 1st floor, and stack was still there. I did buy the house, but,, First I got a surveyor to look at this.In the loft he could tell there was no steel. He said, yes, buy the house, but don't move in until this chimney has been made safe. Naturally, I negotiated a price reduction.
Your situation, chimney on first floor, no obvious support underneath. Yes, it could have been done properly, using a steel. The first question in my mind is, why? It might have been easier to demolish the first floor chimney, then no steel needed, plus you've gained some space in the bedroom.
Second, no paperwork. Doesn't mean it's been done wrongly, because years ago, fewer of us bothered about bits of paper for even quite significant house work. But, it does mean you can't show it was done correctly.
IMO, you've got two choices if you want to sell this house. Accept that it's going to have to be investigated, and if previous work is not structurally safe, then you're going to have to reduce the price to reflect that.Or get it remedied yourself.
But have the buyers told you what their surveyor has reported to them?
It would be quite in order to ask to see the relevant section of their survey, so that both buyer and seller know what has been said, and with what emphasis.
Bit of pain for you. But, tbh, the likelihood of finding any buyer whos not going to want a full investigation is negligible. And now you know it's a potential issue, you have a legal duty to disclose it to any future buyer.
Did you have a survey when you bought? Because if you did, and you weren't informed of the issue, there may be liability there.
Good luck