I've been up 3 times on 2 trips, both September.
First time was from Capilera, up to the top and down the other side towards Granada, with a (hotel) touring load.
The second time was to the top from Pinos Genil, unladen, via Guejar Sierra, and back down the main road, and again the following day, loaded, and over to Capilera and Lanjaron, but without visiting the peak.
Both times were on an audax bike with 28 mm tyres, Conti TT 2000 the first time, Gatorskins the second. I would comment that the ride down to Capilera wasn't a great deal quicker than the ride up (3 hr rather than 4). Such are the problems of unsuitably narrow tyre on a rough track.
On the Granada side the tarmac goes up to the level of the col over to Capilera, but the upper reaches aren't maintained well as it's only Observatory traffic and ski lift maintenance vehicles that normally go past the barrier at Hoya de la Mora (~2500 m). The last 0.5 km or so to the col is unsurfaced, but fairly good, and there's another pole barrier at the col itself.
The road between Guejar Sierra and the main route has been substantially improved since I was there, going by Vuelta coverage. At that time, the road down from the top to Hoya de la Mora, though not good, was mostly OK for 15-20 mph, with occasional slowing for extra rough patches.
Going up the south side from Capilera, it's tarmac for 2-3 km, then a decent gravel surfaced road, leading to a barrier at ~2100 m, at which cars normally stop and after which there's only a bus service up to around 2700 m. Here there's a track going down leftwards to a refugio, the obvious path up to the top of Mulhacen, and the Veleta track contouring up the left side of Mulhacen. A short way up this, there's another barrier. Between this and the barrier at the col at the Pico Veleta, the track isn't maintained, and hasn't been for some long time, so whatever rocks roll down the hill onto it stay there, and it's consequently pretty rough for a road-ish bike.
I wouldn't recommend the south side in June, as it's unlikely that the road would be free of snow (see the current Google aerials, which show how the flat road can accumulate banks of snow that last well past the snow in surrounding areas).