Rugby Borough Council lists 19 conservation areas: Bilton, Bilton Road, Brandon, Brinklow, Churchover, Clifton Road with Hillmorton Road and Whitehall Road, Clifton upon Dunsmore, Coombe Abbey, Dunchurch, Easenhall, Hillmorton Locks, Leamington Hastings, Monks Kirby, Old Brownsover, Rugby School, Rugby Town Centre, Stretton on Dunsmore, Thurlaston and Wolston. Seven of the nineteen sit in or against the town itself.
The council publishes a character appraisal for each area and no separate lime or breathable finish guidance. The borough also has an Article 4 Direction covering small houses in multiple occupation in central Rugby, which is a control on use rather than on materials.
We would explain how you check the conservation boundary, any Article 4 direction and the proposed work before confirming whether permission is needed. That answers it on the render page instead of on site.