The Prime Minister has set out:

Outdoor markets and car showrooms will be able to reopen from 1 June, as soon as they are able to meet the COVID-19 secure guidelines to protect shoppers and workers. As with garden centres, the risk of transmission of the virus is lower in these outdoor and more open spaces. Car showrooms often have significant outdoor space and it is generally easier to apply social distancing.
All other non-essential retail including shops selling clothes, shoes, toys, furniture, books, and electronics, plus tailors, auction houses, photography studios, and indoor markets, will be expected to be able to reopen from 15 June if the Government’s five tests are met and they follow the COVID-19 secure guidelines, giving them three weeks to prepare.

Businesses must be "Covid-19 Secure" to reopen.  Although announced in the Prime Minister's Roadmap, the requirements have not yet been published.

Further details have been promised in advance of the review scheduled for 28th May 2020.