OK these are the building regulations relating to stoves
http://www.planningpo...r/BR_PDF_ADJ_2010.pdf
1.24 says that appliances other than flueless appliances must be connected to flues
I very much doubt your wood burner is flueless!
This and a number of other points put this out of building control
On a practical pont - if this indeed is the case I'd think the biggest risk is that there'd not be enough updraft or it could become blocked or restricted both of which could cause a build up of carbon monoxide.
However I'd not be 100% sure that he's entirely right - perhaps there a flexible flue liner that's become detatched or something like that - definately get a secong opinion.
One thing that does bump up the cost of fitting chimney liners is the need to have scaffolding put up in order to do it - so £1300 doesn't sound totally outrageous if this is included and it depends on the type of liner and whether there is insulation being used here for example.
Mine has a flexible flue with pummice beads insulation and I think it was about that much if I recall