I've been advised I need to drop in here and do a little explaining, and perhaps do a little defense of our kits.
In the 18 years that I've been offering electrical help on TRs, the most common problem by far has been a failure of the turn signals after using the hazard switch. The hazard switch, not being used very often, gets nasty and the TS contacts don't make good connections. The TS power runs through the hazard switch, so when it gets bad, the TS won't work. On my own '71, I took the hazard switch apart and cleaned it twice before I got tired of it and replaced it with a standard toggle switch.
As Randall pointed out, I revised the T\turn signal/hazard wiring on our kits from the standard factory design. Why? Because in order to put the hazard switch in front of the turn signal switch, as the factory did, either a special relay or a special switch was needed. To get these special relays or switches, you either had to buy them from a British parts vendor at a high price, or or buy a hard-to-find relay from an electrical supply house, which could be hard to find. The original turn signal switch is used with our kit, only the hazard switch is replaced. The DPDT switch we supply can be bought almost anywhere.
Now for Dales problem. Given that all four turn signal/hazard bulbs work and flash when the hazard switch is operated, the only area left for the problem is inside the unshaded area in the picture below. Given that neither side works with the turn signals, it is unlikely that both wires from the TS switch to the hazard switch, or the hazard switch itself is bad. The fuse feeding the TS flasher is common to several other items, which I assume are working ok, so we know the fuse is not bad.
That only leaves the flasher itself, the LG/N wire to the TS switch, the connector in the wire, or the the TS switch itself. Check for voltage on the LG/N wire at the switch. If you have voltage here, then the switch is bad. If not, then there is a problem with the flasher or the LG/N wiring.
As to why the flasher is clicking but the lights aren't operating, I'm puzzled. The flasher needs a load to operate, but it doesn't need a full load like the original flashers did. Even a test lamps will operate it.
The bullets/connectors we supply come from British Wiring, and are as original. The bullets are indeed hard to insert, as they should be for a good connection.