It is very difficult to answer this question !
The most pleasant times to begin the pilgrimage in Spain are well understood to be May or September, but these are also by far the most crowded…
Walking in summer can prove to be very difficult in Spain due to the possibility of high temperatures.
There remain the times in spring and autumn which have non-negligible risks of adverse weather, but are often worth the effort, provided one is well equipped! Even then, it will not rain for 28 days !