If you performed N independent tests, any number of which might be probing real or null effects, there is no telling exactly what distribution of p-values you ought to expect.
As you said, if the null was true in all cases, then you would expect a uniform distribution of p-values across your N tests (bearing in mind that in a small sample, the histogram of p-values may show large variability around the population distribution). This is a special case where it is easy to derive the expected distribution of p-values.
If you knew exactly the true population statistics for each test you were doing, then it might also be possible to derive mathematically the expected distribution of p-values for each test, and your overall distribution of p-values across tests would then be a mixture of those distributions. This seems rather complicated to do, but in principle I suppose you could work it out, if you knew the necessary information about the population for each test.
You, presumably, do not know the ground truth for each and every one of your tests (or else you wouldn't be doing them in the first place). Therefore, I don't think you can form a strong expectation about the distribution of p-values you ought to see, other than: if for some portion of my tests, the null hypothesis in truth does not hold, then across all my tests I expect to reject the null more than 5% of the time (assuming p<0.05 is your criterion for rejecting the null).