Today is the day the 5th Money Laundering Directive required individual nations to have implemented the 5th Money Laundering Directive. European directives require legislation in each nation whereas European regulations do not, they are directly enforceable. And don't think that in leaving the EU we won't still comply. Firstly because in order to work with Europe they will require the same sort of standards but also because it was the uK who was pressing for better money laundering rules in the first place!
However, in reality, for the lettings market this really is a "non event". Don't be fooled by reading about letting agents have to register for money laundering and do due diligence on their clients. Indeed this is what the law says, but you have to read the definition of letting agent and this is where it only applies to a letting agent who is letting a property in excess of 10,000 Euros a month (about £8,300). This will therefore be a tiny percentage of properties and agents. Searching Rightmove for the whole London area and properties over £8,000 a month today (10/1/20) it finds 2,797 properties listed over £8,000 a month. Outside London you find far fewer. The point is that you are not considered to be a letting agent if the rent is under 10,000 Euro.
For some agents they may already be registered for sales and any money laundering compliance will be a small step. For the vast majority they will not need to do anything different as they simply don't rent any properties over the 10,000 Euro threshold.