When I create a Relative Date Slicer on Power BI Desktop it works, i.e. if I click on the drop down for Last / Next / This I can choose something. When I publish that and access the report via my browser, while I can click on the drop and and see the value, I cannot select Last / Next / This. The same applies to the drop down for Weeks / Months / etc.
My current workaround is to click in the numeric field in the middle, tab backwards and forwards between drop downs, and use Alt + down then the arrow keys to select what I need. Well, that and remove them from reports because explaining that to users is a waste of time.
My Power BI desktop version is: 2.59.5135.601 64-bit (June 2018) (not that this probably matters as the drop downs work on Desktop).
The Power BI online version this is happening on is: 13.0.6700.151