Then your wife should contact a nearest amphor and ask how she can register her marriage. Sometimes different amphor require different documents for updating registration of family status. About your visa, it’s better to ask immigration office that you plan to extend your visa. In our case is Chiang Mai, I went to the office and asked them by myself.
If she is willing to stay in Free State Quarantine, then she has to wait for new quota approval from Thai government. Free quarantine for Thais traveling from London by Thai Airways is now approved until February 7th.
Better start searching info asap, I'm not sure if you could make it to Thailand before Christmas or not. I called RTE The Hague last Friday to request an appointment for visa application, the earliest day I could reserve was for December 18th. Please note that you have to quarantine at an ASQ hotel for 15 nights before getting to see your girlfriend. More information and procedures are available on RTE The Hague website.