If interest rates go up, bonds get sold (for better yield bearing products), pushing the yields of those bonds higher. And it finds some equilibrium. The fact it isn't immediate has to do with short term vs long term bonds. When they mature and the pace of arbitrage.
I don't see how a rate hike is meant to lower mortgage rate. And just looking at the figures shows it's the opposite effect.
Logically, if borrowing money becomes more expensive, how could borrowing specifically for the purpose of buying houses become cheaper.