After removing the boarding from the base of the chimney I found it
straight
but capped.

Then it will be possible, but possibly not cheap or easy to find a plumber
willing to do it. It may be a scaffolding job to fit the top end safely.

Again, do the sums to see if a standard concentric vertical flue (p. cheaper
boiler, but more expensive flue) or a drainpipe flue system will be cheaper.

Christian.