It's a scruffy looking job at best with missing nails (especially up the side and base of the timbers), nails not hit fully home, split laths, ragged lath ends, etc... have you paid them in full ?
May be worth taking a bolster chisel to one if you can as the mortar may well come off easier than you think. Failing that, then yes, it is possible to bed them as they are found but would be much more work getting them set and to a level, etc...
What you have shown there is a lap joint but very much the way to tackle this one. You still actually have around half of the timber depth at the base so using modern foaming glues and good screws you could cut out the rot with a multitool type weapon and patch in a new piece of timber and if...
The upper short diagonal valley is the main culprit i'd say as the water will be running at speed down from the higher pitch and running under the tiles as there is no pointing in the valley. The vertical section of lead up the main roof should also be under the next tile above and not stop mid...
Nice handles and fitted to the very best cabinets and furniture in the 1800's / early 1900's and the piece you have was probably quite a sum of money back then. Anyway I found some on here for you...
https://www.ebay.co.uk/itm/355576942674
These are a good alternative as they extend to the space available and tighten up with a quick twist :
https://www.dunelm.com/product/bathroom-basics-extendable-4-tier-corner-caddy-1000030058?defaultSkuId=30110114&branchCode=0480&gStoreCode=480
You need a firm surface so scrape off anything that is loose or flaking with a scraper and wire brush and give it a coat of PVA and then you can give it a coat of Bonding to level it up ready for the backer boards.
If it was built in the 80's you should be able to go on your local council Planning Dept web site and search for the address which may give you the planning application details with the drawings etc. While that may not give you full structural details it will give you the applicants name and...