You are right to be concerned with changing the pH. Some of our beloved trees would not grow too well in very acidic soil!
I researched this because I had similar concerns. After organic material degrades and becomes compost if has close to a neutral pH. Before then, it might change the pH locally where it interacts with the soil, but it is a small enough effect as to hardly matter, especially when it comes to established trees.
Significant change really can only come through concerted effort-adding limestone, sulfur, etc-and even then, you need to re-apply every few years. The only time I would look very closely at my mulch for pH would be if I am germinating seeds or have tiny seedlings.