The hoop backs are steam bent beech, so DIY replacement will be all but impossible, however, you could try making up curved cauls to match the curvature of the back where the damage is, dripping a liquid epoxy resin (such as West System) into the cracks, then cramping with cast iron G-clamps and leaving for 24 hours.
The wooden cauls would need to be heavily waxed to ensure release and they will need to be a good fit to the curves. You will need proper cast iron G- cramps to achieve the clamping pressure required. The only glue with sufficient strength will be an epoxy
As @foxhole says, not an easy DIY repair. Or an easy pro repair for that matter
If you need to find a tradesperson to get your job done, please try our local search below,
or if you are doing it yourself you can find suppliers local to you.
Select the supplier or trade you require, enter your location to begin your search.
Please select a service and enter a location to continue...
Are you a trade or supplier? You can create your listing free at DIYnot Local