Using a mechanical switch on output of SR latch / OR gate?
Yeah, so a signal with a 10 ns rise time will have, if you want to keep it similarly sharp have to have at least 5×, better 7× or more bandwidth than 1/(rise time). This rule of thumb comes from the series expansion of the square wave of frequency 1/(rise time).
You will probably have not much trouble with the cleanliness of your edge even if you use a big chunky switch, but you'll radiate stuff like crazy.
On the other hand, the soldering is clearly overkill: Switches meant for such bandwidths do exist. For example:

Use your manual switches to control the SEL inputs.
This specific device probably won't fit your voltage range requirements (you don't specify). You got to pick one multiplexer fit for your application – there's literally hundreds to choose from at the distributor's websites (Farnell/element14, digikey, rs components...).
In all honesty, though, it might be simpler to simply have amplifiers / buffers that you switch to a high-Z output mode.  In all likelihood, you could even use mature, but advanced 74xx logic like the SN74AUP1T157; nothing too fancy.
General recommendation: Make sure you really need the 10 ns rise time. While far from impossible, I wonder whether a Schmitt trigger on the gate driver (integrated or external but geometrically close) wouldn't relax your bandwidth requirements significantly and thus allow you to increase rise time, making your system less prone to EMI.