Are there lockers or left luggage at Bath, Oxford, and Stratford-on-Avon?
According to the National Rail site, there are no left luggage facilities at the Bath Spa, Oxford, or Stratford-upon-Avon train stations.
You can store luggage at the Funky Hostels close by the Bath Spa and Oxford stations. See here.
Sometimes hotels will allow people who are not staying there to store luggage for the day--for a fee. You could send an e-mail to the Stratford-upon-Avon tourist office and ask if they know of hotels that would help you out. See this page.