I can't seem to get a straight answer on this. Can you work if you are a stateless person and file for residency? They give you identification paperwork, so I don't see why not. But.......
If you are outside the UK you need a visa to enter for work. You can check this out here https://www.gov.uk/check-uk-visa/y, by selecting ‘Stateless or refugee’ as the nationality option.
If you’re in the UK you have to apply to remain https://www.gov.uk/stay-in-uk-stateless and https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/501509/Statelessness_AI_v2.0__EXT_.pdf Section 6 states that those granted leave to remain are entitled to take employment and have access to public funds