Mild,
I had the exact same problem. LED rear lights. When I put in LED front indicators, they dont work and from memory the rears stop working also when the headlights are on.
Only difference is that with mine all is OK when I put the normal bulbs back in the front indicators.
You should check the earth on the fronts.
If that doesn't fix it - search on my posts for a thread about it. I went to great lengths and the thread explains how the indicators are wired and so should help you troubleshoot it.
Did my head in and gave up and just left the normal bulbs in the front.
Cheers,
Frank.