The Fallout 4 engine is old, I think it's a modified version of the Creation Engine which powered Skyrim, which is a modified version of the Gamebryo Engine from Fallout 3. In other words, you don't need a super high-end PC to run the game. If you don't need this PC right now, you can wait until Q1 2017 when AMD's Zen CPU's and Vega GPU's are out which may, hopefully, put downward pressure on CPU and GPU pricing.
An LGA 2011 CPU and quality motherboard is still going to be expensive, you may want to budget a bit more than $1100 if you want to go that route. The CPU and motherboard for an LGA 2011 would easily be over $500.
Don't worry too much about full DirectX 12 support, games are slowly being made for it and it will still be a while before it is commonplace. The RX 480 is a nice card, but hard to find. A GTX 1070 is more than adequate.