Is that on the ground floor? Trying to make out what is below the joist.
Having a similar situation, but not as bad as yours, I cut out a section 200mm long and 15mm section out of the top of the joist, well trimmed the joist to that size as that was the bit to get rot free wood and then replaced the bit with fresh wood Then made the area in to a flitched joint by adding a metal plate and similar sized wooden joist bolted through. https://en.wikipedia.org/wiki/Flitch_beam
Is that on the ground floor? Trying to make out what is below the joist.
Having a similar situation, but not as bad as yours, I cut out a section 200mm long and 15mm section out of the top of the joist, well trimmed the joist to that size as that was the bit to get rot free wood and then replaced the bit with fresh wood Then made the area in to a flitched joint by adding a metal plate and similar sized wooden joist bolted through. https://en.wikipedia.org/wiki/Flitch_beam
Or give these people a shout http://www.joist-repair.co.uk/