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If your looking to do this you will need building control approval, it would be cheaper to get a heating engineer in to do the work, otherwise your looking at months to get the required approval and hours of calculations and documentation to submit to self fit.
The size of the dual coil is relative to the BCU output you need to provide, the heat output of the solid stove, heat output of the oil fired heater, and flow rate of the system.
Logically both systems could be sealed\pressurised systems, you just need to locate the expansion tank where both the oil and stove can benefit from them, your also looking at having to have a rather complex multi port valve to control the flow as you would probably not want to have both systems running sanctimoniously.
So I can close the case I would appreciate it, if you can rate the service received using the star rating on the question page.