Where to look?
Age UK (the UK's leading aged charity) has a network of 130 locations across the country. 70 of these offices provide a paid handyman service. Age UK handypeople are taught not just to perform regular home maintenance tasks but also to keep an eye out for additional small jobs that could make the home more comfortable (for example, heat-proofing). Use the postcode search tool to learn more about what's offered nearby.
For those living in London (within the M25), Silver Saints is a handyman service that attained key worker status during the pandemic. Services can be booked online, and you can choose between same-day and next-day options.
Use the Check-a-trade website. Check-a-trade tradespeople have been vetted and checked prior to being listed on their site. Reviews and customer feedback are extensive.
The Elderly Accommodation Council (EAC) lists handy/odd job persons across the UK on their website. This is a directory so you will need to complete your own reference checks.