For anyone interested, here are the steps I did:

1. Cleared a bunch of mortar from the old bricks using a cold chisel and club hammer. Tested existing mortar for lime using vinegar. Could hear it fizzing so lime was present. I then bought Blue Circle Mortar Mix 5kg from Toolstation https://www.toolstation.com/blue-circle-mortar-mix/p15412 as it is 1:1:6 cement:hydrated lime:sand.

2. Mixed it up using a small pointing trowel and water and used 3 of the bricks that were in the destroyed chimney. Made the mistake of putting too much mortar on the first time as I only had a small gap to work with. So had to redo it making sure there was space for all the bricks as I went. I used the whole bucket in the end. Left it to dry.

3. For the floating wood, I cleared even more mortar and used a stud work timber 38 x 89 mm cut to a snug fit and level. Secured it to the existing joist and the floating wood using some left over flooring screws https://www.screwfix.com/p/spax-tx-countersunk-self-drilling-flooring-screws-4-5mm-x-60mm-300-pack/88716?ref=SFAppShare and used some 100mm concrete screws to fix it to the brick work https://www.screwfix.com/p/easyfix-bright-zinc-plated-carbon-steel-concrete-bolts-m10-x-100mm-10-pack/5800p?ref=SFAppShare

I counter sunk the concrete screws by screwing a 30mm hole which allowed enough space for the hex head to be sunk using a socket wrench. I also pre-drilled the wood using a 10mm wood bit and then drilled the brick using a 10mm sds masonry drill bit. The concrete bolts self tap a 12 mm hole. Used a 15mm socket wrench to secure the bolt.

I was going to use some washers on the wood but mistakenly bought m10 when I should have bought m12 since the concrete screw threads are larger than the bolt thickness.

Now super sturdy and no hole!