On December 8th 2019, Farage said in an interview with Sky News that after the UK leaves the European Union (as occurred on the 31st January 2020), the Brexit Party would change its name to the 'Reform Party', and would campaign "for changes to the voting system and the abolition of the House of Lords".
In the same interview, he said that he had already registered the name. However, a search of the Electoral Commission records returns no trace of such a party having been registered by Farage or any of his associates. In addition, it does not appear to have been rejected, and is not listed in the record of current applications either. A search of Companies House does return "The Reform Party Limited", but this company was registered after the interview, and none of the directors include Farage or any of his associates. In contrast, Farage is still a director of The Brexit Party Limited.
So in conclusion, according to Nigel Farage, the new party will be called 'The Reform Party', and has already been registered. However it is unclear what Farage meant by 'registered', as neither Companies House nor the Electoral Commission seem to have any record of the registration.
November 2020 Update:
At the beginning of November, Brexit Party chairman Richard Tice along with leader Nigel Farage published an article in the Telegraph, stating that they would seek to rename the party to "Reform UK". Details of the application are not yet available on the Electoral Commission's website, however, the Telegraph also published comments from the commission:
On Sunday night a spokesman for the Electoral Commission said: "The
Brexit Party has submitted an application to change its name to
'Reform UK'. If we are satisfied the application contains all of the
information required by law, it will be published online. The
application has yet to be assessed, this may take up to six weeks.
"Applications to register a new political party name are subject to
detailed assessment against the criteria set out in law. When we
receive an application to register a new party name, the details of
the new name are published online so members of the public can input