The lake at Attabad continues to fill quite rapidly, with the water level now rising at close to 1 metre per day, as this graph using Focus data shows:

The remaining freeboard is now 15.5 metres.  At the current rate of increase, water will flow across the spillway on about 22nd May.  However, Young and Hewitt (1990) provided a hydrograph of flow in the Hunza (apologies for the poor quality of the image) and rivers downstream:

The data for the Hunza are from Dainyor, which is downstream of Attabad at the point at which the Hunza joins the Indus near to Gilgit.  Thus, the recorded flow should be higher than at Attabad.  However, the key point is that the June discharge is about five times that of May.  Thus, we might reasonably expect to see a rapid rise in inflow over the next few days, which may bring forward the date upon which water flows across the spillway.  Meanwhile, the most recent Focus weekly update contains the following two images of the current state of the spillway:

Reference:
G.J. Young and K. Hewitt, 1990.  Hydrology research in the upper Indus basin, Karakorum Himalaya, Pakistan. In: L. Molnar, Editor, Hydrology of Mountainous AreasIAHS Publ. vol. 190, Wallingford, UK pp. 139–152.  Available here.