I can't answer that with certainty but my guess is, yes, the roads should be open, but the longer you wait the shorter the window of opportunity remain open? I say that because many years ago (I think it was 1990) I took a trip to Naran (then a hike to Lake Saiful Mulook)--probably from Abottabad to Kwai to Naran.
That was in April.
Things were still frozen in many places. I had very little chance to find someone to take me from Kwai to Shogran. But I wanted to see Shogran and Sri Paye. The rest of my party--I was part of a scientific event in Ayyubia--decided that it was not worth even trying to go to Lake Saiful Mulook because of high probably of snow/ice. I was not going to give up. So I broke away from the team--which went back to Karachi. I was alone after that.
I think I was dropped off on the main road between Mansehra (?) and Naran at Kwai. At Kwai it would be very hard to find someone to take me up to Shogran. So I decided to hike the 8 KM up. And I made it before dark and checked into a small hotel. Shogran was almost empty.
The next day I hiked to Sri Paye. I am not entirely sure if I made it all the way up or not to the lake or not--there was good bit of icy walk. In my lonely hike I ran into three American women hiking to Siri Paye. We had a nice chat. Silly me--a Karachiite--trying to impress these women from Wisconsin or Minnesota by telling them how to walk on icy trails!
Anyway, not to spoil your thread here but I hope this helps. It's been 23 years but I remember Shogran to be one of the most beautiful places I have ever been. And I am pretty sure I made it to Siri Paye lake as well and the view was gorgeous there--I think one can see Kashmir from that location.
Enjoy your travel.