A sweep* turned up to do our chimney last year, but wouldn’t touch it because it wasn’t lined. So we got it done (along with another that we rarely use) £1800, which was fair enough; at least 4 blokes, all day, including ascending our very steep roof to drop the liners in, and removal of bricks to accommodate.
*A cunt, who could have told us he wouldn’t sweep an unlined chimney before he arrived and charged us a call out fee.