Not an expert, and I don't think this is a factor at all, but the cold showers are in part meant to help you tap into a natural ability to stay warm (without shivering), so you're not get an appreciable dip in core body temperature in the first place if you do it right.

I believe it is thought that long term Tau accumulation in the brain is what worsens or accelerates alzheimers in patients, but a healthy person is clearing Tau build-up all the time, primarily during sleep. Any behaviour that improves sleep quality is therefore likely to be a boost in preventing and decelerating the development of alzheimers and cold showers promote good sleep and help keep our circadian rhythms aligned in a way that encourages good quality sleep.

One of the reasons why a hot bath before bed is good for sleep is because it decreases body temperature.