From Humanists UK website:
In July 2020, in a case brought by six humanist couples, High Court judge Mrs Justice Eady DBE ruled that the failure to provide legally recognised humanist marriages means that ‘the present law gives rise to… discrimination’. She also ruled that, in light of that, the Secretary of State for Justice ‘cannot… simply sit on his hands’ and do nothing. However, she said, given that the Government is currently giving the matter consideration in the form of a review into marriage law by the Law Commission, the Government’s refusal to act immediately can be justified ‘at this time’ and concluded, ‘Although I may deprecate the delay that has occurred since 2015, I cannot ignore the fact that there is currently an on-going review of the law of marriage in this country.’
In response to an email, local MP for the St Ives constituency, Derek Thomas, wrote:
“The Law Commission conducted an initial scoping review of marriage law. It found that the law was badly in need of general reform and that it would not be appropriate to legislate solely for non-religious belief organisations, as this would create further anomalies. The results of the Law Commission’s consultation is finally due to be reported to Government in July of this year.”
So it seems that progress may come, as a result of the Law Commission’s work, when it is published in July 2022.