If you were already married and wanted a ceremony, you’d have to get someone - be that a humanist minister or whoever to do fulfill that role but I wouldn’t have thought that was too difficult. Given there would be no legal function of the ceremony literally anybody could do it.
My friends have done precisely that - they've had friends of theirs as celebrants, at ceremonies in the UK and abroad. My cousin also had this for their wedding - a mate of theirs was the celebrant and just served as master of ceremonies to invite him and his (already legally married by signing a marriage register at the registry office) wife's vows, and readings from their friends.
(and it's not just, or even primarily, Muslims: it's a huge problem in parts of the Jewish community as well
On a recent C4 documentary (I think it was Dispatches) called 'The Truth about Muslim marriage', a contrast was clearly drawn between the fact that Imams are not required to be registered to conduct legal marriages / the number of Muslim wedding ceremonies which take place in the home, with the requirement that Christian and Jewish weddings are required to be legal marriages as well
That said, in order for a rabbi to conduct a legal wedding, the marriage must be in accordance with Jewish law - the celebrant for our wedding was a Rabbi who officiated at interfaith / Jewish secular weddings, and he was very very clear that he could not legally marry us because it was not a Jewish wedding, and that my husband and I could not be married in accordance with Jewish law because I am Jewish (under Jewish law) and he is not.
There are most certainly issues with women needing to get a Jewish divorce (known as a 'get') in addition to a civil divorce in order for the marriage to be considered dissolved under Jewish law (and for the person to get remarried under Jewish law).
However, unlike in the Muslim community, it isn't a widespread issue that a large number of couples believe themselves to be legally married when in fact they are not. A Jewish wedding has to be a civil wedding in addition. If it's not, it's explicitly not a Jewish wedding. A liberal Rabbi may perform an interfaith blessing, or act as a celebrant for a ceremony like ours, but it cannot be a Jewish wedding unless it's in accordance with Jewish law. In these cases the Rabbis make it explicitly clear that these ceremonies are neither considered marriages under Jewish or British law - they are a celebration only, and the couple must have a civil marriage, conducted by a licensed registrar, to be married in the eyes of the law. This Guardian article summarises the key themes from the programme
"Three out of five of the British Muslim women surveyed did not, in addition to their nikah ceremonies, have a civil marriage, rendering them outside the legal protections and provisions that marriage brings...In the documentary, Rabbi Herschel Gluck described British Jews benefiting from a marriage “package deal”, going on to say: “When you get married in Jewish law, you’re also getting married in civil law."