The Office for National Statistics (ONS) publishes a country and region breakdown of R&D spending in table 6 of the annual release on UK gross domestic expenditure on research and development (GERD). The aggregate figures for 2018 were:
Yorkshire and the Humber
The ONS does not publish a breakdown of R&D spending by local authority area.
HM Revenue and Customs publish a breakdown of corporate R&D tax credit claims by county and unitary authority area at https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/878879/200408-_R_D_Supplementary_final_tables.xlsx, though those numbers are distorted by headquarters reporting effects, since the local and regional allocation is based on the postcode of the company's registered address, which might not correspond to where the R&D activity takes place.