so do you really want to do this “wirelessly” ? If so, that’s a much harder task.
First up, none of this will be Gadgeteer module specific. Far too much “infrastructure” with PCBs and extra space consumed that you don’t have available. You’re going to have to go back to individual components and wires and the like.
Buttons. Sure, low-profile buttons are available and can be used under grips or tape. You’d need to find one that suits, but a button is about as simple a device as you can find 
From commodity suppliers, an example is https://www.sparkfun.com/products/8720 or https://www.sparkfun.com/products/97 but both will require soldered wires onto the pins and may have varying degrees of waterproofness…
LEDs. Depending on how fancy you wanted to be, you could custom make some quick-and-simple PCBs and mount LEDs to them, or you could buy some PCB mounted LEDs and wire them up. Depends on how fancy you wanted to be with colours, flashing, and re-using LEDs for multiple purposes whether something like the bars they talk about in https://learn.sparkfun.com/tutorials/led-light-bar-hookup might suit or if a custom design is a better idea. Waterproofness again will be your challenge.
Wireless. Easy off the shelf options are available, like zigbee. Other cheaper options are also available. None of them are likely to result in a “small” package. I’d consider a wired solution here, doesn’t cost anything like a wireless solution might (and the biggest cost will likely be integrating a wireless module in the solution, in space and time, and a wire is just quick and easy). If you went with say zigbee or some other “transparent pipe” networked device, you could simply use two general IO pins to detect when a button is pressed and send that back to the other end; when both are pressed, send the status back at the same time, don’t try to differentiate.
Wired: I would run a single four-core wire (maybe just split off the required cores from something like this https://www.sparkfun.com/products/10648 ) that separates at the stem and goes to the left/right buttons. You could use 3-cores and share a GND wire between the two, but using 4-cores with two going left and two going right seems most sensible to me. You could use the same approach as the standard cabling methods for older speedos when they used to run the sensor wire for cadence pedal sensors back onto the frame via the brake cable itself. (I know, less than “clean” especially on a nice looking bike like yours is)
Processor. Depending on which way you approach it (wireless between front/back, or wired) might change the processing requirement, but a small micro would suffice (it is unlikely you’d need it to be anything powerful or running netmf, but you certainly could use a Cerb40 for instance).
If I was doing this: I’d make a PCB the way I wanted to for my LED bars. I’d probably have WS2812 multicolour LEDs on a small PCB that was shaped like a double-ended arrow, and have it flash orange arrow pointing left when turning left, flash orange arrow pointing right when going right, and flash a red bar when stopping. That could be done with a Cerb40, so I’d probably include the footprint for Cerb40 on the board to make it simple. Then I’d run wires to the buttons for simplicity.
