Yes they should have details of when you worked for them so just ask.
Why do you need it ? very few new employers need more than the rough dates. I have had over 20 jobs and some of them I can't even remmber the year let alone the start date but it has never been a problem.
No I don't have P60s. I need to have traceable employment history for the past 20 years. I can do the last 15 yrs but struggling beyond that. Previous employment in catering and all have moved on, no longer in business or deceased.
Well if you don't know the exact dates I don't see how a potential employer can know whether what you say is correct are not. Many of the firms from 20+ years ago will have moved/merged/closed down or have no record of you so an employer is very unlikely to be able to check so far back