I don't know your level of expertise so please don't be offended by my reply. Also, don't take my reply to imply I necessarily know a whole lot more about electrical systems than what I say here!
One of the unfortunate realities of owning old cars is the existence of DPOs (dreaded previous owner realizing that I will be the DPO to whoever buys my car). They love to monkey with the wiring to the consternation of subsequent owners. Judging from your pictures, we have hope that yours is not mangled since your wire colors are correct and you still have the wiring tape on some of your wires. a DPO would not likely use wiring tape.
You are using a non standard switch so you'll need to "translate" your switch to the stock switch - in other words, wire it so it does the same thing. Judging from your initial post, it appears that you understand that. I assume the relay you mentioned is the DB10 unit that is labeled "RELAY" in the attached wiring diagram snippet. It is a rectangular box about 4"x2" behind the dash.
Do your turn signals (indicator lights) work correctly - they flash at the proper time? If so, it could be that your hazard flasher doesn't work. You can swap your indicator flasher for your hazard flasher to test that. If your problem moves with the flasher, that's your problem. If not, you can probably ignore (most of) the rest of this post as it sounds like your switch is wired incorrectly. In either case, bookmark
http://lotus-europa.com/manuals/ as it has a lot of essential as well as interesting information about your Europa including searchable manuals and color wiring diagrams.
Kendo's suggestion that you may have LED lights in your turn signals is another thing to check. What he meant to say, I think, is that stock flashers (which you have as shown in your third picture with the purple wire) are mechanical and work with a bimetallic strip. The current draw from the turn signal's incandescent bulb heats up the strip causing it to bend and break the connection, the strip cools and bends back causing it to hit the contact which is the clicking sound you hear, the turns signal lights light up again which causes the bimetallic switch to heat up again and the process continues... LEDs don't draw enough current to heat up the strip so you either need to add resisters inline with your turn signal lights or get an electronic flasher (I recommend the former. Electronic flashers are either silent or they beep rather than click. I've gotten used to the beep!
). They are usually available from automotive LED light suppliers. So as Kendo suggests, if you have LEDs, you won't get flashing but your lights should just stay on. That's easy enough to confirm by opening up your turn signal lamps and looking at the "bulb."
Kendo also makes a good point about your relay - that it might not be stock and you may have a DPO problem (wiring that may need to be corrected) - but I don't think the relay will affect the current across the flasher as the relay will be either switched or not (I'm happy to be convinced otherwise, see second sentence of this post). Solid state DB10 replacements are available from Moss Motors and likely any parts supplier for older (British) cars.
Hopefully that helps! If it doesn't, maybe you can supply more symptoms or evidence.
Good luck!
