The Dublin III Regulations that provided for asylum applicants to be returned to the first EU country they entered no longer apply post-Brexit.
The new rules are complex and appear to rely on non-existent international agreements with other countries to accept asylum applicants from the UK.
There is also published guidance:
https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/947897/inadmissibility-guidance-v5.0ext.pdf
There are likely to be legal challenges about this regime, including on the basis of the lengthy delays it will mean to the already very slow asylum process and in particular the impact of any admissibility process for those in immigration detention; vulnerable applicants such as victims of torture and those with mental health issues as well as victims of trafficking.
Leaving the EU does not change the UK’s obligations to offer protection to refugees as a signatory to the 1951 Refugee Convention. In fact the Government reaffirmed its commitment to the Refugee Convention and said that the UK’s status as a “world leader” in the field of asylum will not change once it is no longer subject to EU laws specifying minimum asylum standards.