Posting a real annotated schematic, not a frizzy picture as you have wired it will help as will links to "Technical" information on the hardware. Be sure to show all power connections and supplies.
To me, "fairy lights" indicates a short string of LED lights with a "battery box" with a few AA or AAA sized batteries and a mode control. Modes may include things like light intensity settings and/or blink modes. The modes are typically selected by a push button wherein the mode control sequences (e.g. ... on high -> on low -> blink -> off -> ...). So fairy lights also have infrared remote control.
As such the problem comes down to simulating button pushes on the battery box. This is a common problem in Arduino projects interfacing with off-the-shelf electronics, but the particular solution comes down to electrical characteristics of the particular button. Thus one may have to experiment to figure out a suitable way to "push the button" and that's the hard bit. Timing itself is trivial, interfacing to some undocumented piece of electronics may not be trivial.