Hope for sunshine with +3 in the morning, and +15 during the day. Prepare for rain, wind, and snow at higher elevations with -5 in the morning and just +8 during the day. If you ask me again on March 21st, I should be able to narrow it down for you a little bit more.
What @C clearly said. It's the perfect answer.
I'm one of the oddballs who think March is the best month to walk. But as far as weather is concern...anything can happen, and maybe all in one day. So be prepared for anything from snow and sleet and freezing rain to balmy sunny days. Wind and rain are pretty much guaranteed.
Just out of Roncesvalles this year. That night we had 25cm of snow in Zubiri. They year before it was probably sunny! Who knows what it will be like. This year in March a cold snap in Europe hence all the snow.
Rely on layering rather than overly heavy single-purpose outer layers. On a windy day, a poncho makes an effective wind barrier, even if it is not raining or snowing.
Concentrate on keeping your head, hands and feet, warm and dry, as a priority. Add or strip off layers as needed.
I recommend neon-colored poly runner's gloves and a beanie hat. I use a poly fleece cap, and microfiber gloves, in neon lime-yellow. The poly insulates even when wet and dries very fast.
Plus, the loud color contrasts against your surroundings and helps raise the safety margin when you have to walk on a road. The Valcarlos Route from St. Jean Pied de Port has significant road walking. That is why this route remains open all year, except for the heaviest snowfalls. They need to keep the road open...
I walked the Camino in 2015 from March 23 arriving in SdC on April 23rd. Overall I thought it was a great time of year to walk. Yes there were some days of cold, snow, rain as you can see from my picture on March 23, 2015 my first day SjPdP to Roncesvalles on the low route but after that first day saw little or no snow. And many more beautiful spring days than bad ones. I will admit I come from a cold, snowy climate so my training walks in Jan-Feb were in snow/cold but as long as you have the proper clothing layers all will be good! Buen Camino!
Prepare for everything. We saw several days of snow, sleet, hail and many more days of rain and muddiness (if that is a word) temperatures several days of lows below freezing and highs up to 60 f. Only a total of 6 days of sunny weather. That was 2 years ago, and I would do it again!
I have been astonishing lucky with early April start dates on the Camino for four trips. Probably only had my rain poncho on six times over all for trips combined and only for brief showers. I have never had to walk a full day in the rain in for April starts and one mid September start. The only year we ran in to snow was in 2016 when we left SJPP on April 8th. Raining when we left, but even on the low route to Val Carlos we were in pretty heavy snow the last couple of hours into Roncesvalles. The following day the snow was pretty much gone by the time we got to Zubri.
After that the weather was pretty good for the balance of our walk.
Walking in snow is not a big deal as in northern Alberta we do it for six months a year.
As others have stated is the only thing guaranteed is that the weather is changeable and one year's experience is essentially meaningless as to what to expect the next year.. In 2014 I was in short sleeves and shorts almost every day before noon as early April was a warm spring that year.
The following year it was much cooler.
Be prepared for changeable conditions, but don't get stressed out about it.
We left SJPdP on March 15/18 we started with a gorgeous day then from Valcarlos it was lovely in the morning but snow in the afternoon from then on we didn't really have great weather, wind rain snow with some breaks. But no complaints I frankly loved every moment of it. Better weather was found closer to Santiago