I think that the problem lies in the blinker idiot light. It "sits" between both switch poles. When you turn the switch either way, it gives +12V to the corresponding set of blinkers, and also to the idiot light. Then, the idiot light is grounded through the other set of blinkers. The current running through it is high enough to lit the idiot light, but low enough to not lit the opposed set of blinkers.
My guess is that by using LED blinkers, that need much less current, the current through the idiot light is high enough to lit the blinkers.
How to diagnose? Remove -or disconnect- the blinker idiot light and try again. This time it should work as expected.
If you still want to go with the LED conversion, you will need to modify the wiring. Either use a blinker relay that have a separate connector for the idiot light, or rewire the idiot light, taking one of the poles to ground, and the other to both switch poles via diodes.