"Sistering" the joists as he proposes is a standard solution and probably the surest way but perhaps worth checking why they sagged when the floor is lifted. Could be they were just too small from the outset or the ends have rotted or a support mid span has settled. The cause might suggest another solution. If the ends have rotted then perhaps investigate the cause.

If this is a first floor flat then the new sister joists probably shouldn't be the same depth as the originals eg there should be a gap between the underside of the new joist and the plaster - this is to stop cracking of the plaster downstairs as the load comes back on the new floor.

Also if this is a first floor flat consider being a good neighbour and adding sound insulation...but watch out if they have downlights downstairs.

I don’t know if its worth considering using 18mm OSB to laminate your existing joists. It may well be more economical than using sister joists. Have a look at this guide  regarding the repair of floor after over doing the notching, it will give you the idea.  

Many thanks for the advice.  It's good to know that 'sistering' joists is standard practice.
I am in the 2nd floor so will put some sound proofing down, and I do have cables running through the existing joists so this will need some thought.
Thanks for taking the time to reply.
sr