In order to try and take advantage of the longer hours of daylight it is advisable to do the challenge in June-August. July in particular can be very busy with many charity teams doing the three peaks, expecially around the summer equinox.
It is also best to avoid any bank holidays as extra traffic on the roads will cost you valuable time. We decided to try and do the challenge in August so that we still had relatively long daylight hours, but avoided the busier July weekends.

This is fairly crucial to the whole challenge and you will have to have a good idea of an overall schedule, along with approximate target times for each mountain and for waypoints along the route. I have set out below the target times for each leg of the challenge, and these are maximum times in order to ensure you get to the end by 23hrs and 30mins.
Ben Nevis
3 hours
2 hours
Driving time to Wasdale Head
6 hours
Scafell Pike
2 ¼ hours
1 ¾ hours
Driving Time to Pen-y-Pass
4 ½ hours
2 ¼ hours
1 ¾ hours
23 ½ hours
Be careful when planning your driving routes as programs like Autoroute or Google Maps can seriously underestimate the time taken on the smaller roads around Wasdale Head. The times above are inclusive of any stops you make en route at service stations.

This schedule is realistically as slow as you can go, and hopefully you should be able to do each mountain faster than the above times.

Start Time

The other issue is your start time and there are two realistic options, either stating Ben Nevis at around 6am or around 5pm. I have listed the advantages and disadvantages below:

Early Start
Late Start
Scafell Pike hardest to navigate so climbed in day
Most of Scafell Pike climbed in dark - most difficult of the three to navigate
Finish at 5-6am so nowhere to go and celebrate
Good finish time so can go to pub afterwards
No hanging around in Fort William
Get to sleep during night drive
Roads may be busier coming through Scotland and to Wasdale Head
Arrive at Wasdale Head very early morning - complaints from local residents about noise
Empty roads heading in to North Wales
Quiet roads through Scotland and to Wasdale Head as driving through night
Snowdon done in dark but is easy to navigate
Easier parking at Wasdale Head as no tourist traffic early morning
Awake for shorter period of time - not awake all day before starting
Ben Nevis may be quieter as no tourists in evening
Can park at Snowdon due to very early arrival
Busier on Snowdon as you summit at peak time of around 1pm

In the end we decided to go for the late start as we felt it suited us better. We felt that there was a large advantage in driving through Scotland and to the Lake District during the night as the roads would be relatively quiet. It also meant that we would be ready to sleep in the car on this part of the drive having been up all day and just finished Ben Nevis, and this was easier during the night rather than trying to sleep in the daytime.

One big problem for the late start are that Scafell Pike would mostly be climbed in the dark after arriving at Wasdale Head around 3-4am. Scafell Pike is the hardest to navigate as the rocky path can become very indistinct as you get towards the summit. However, having done a training walk on Scafell Pike we felt that we should be able to cope with this issue. The other big problem would be if the weather was particulalry warm in Wales on Sunday as we would be climbing Snowdon around midday. Our second training walk was an example of this when we had blue skies and warm weather which made the climb harder. We decided that if the weather was looking like it would be hot on Sunday we would try and start Ben Nevis a bit earlier in order to hopefully get to Snowdon earlier in the day and avoid the worst of the heat.

Another reason for starting late is that if you stay in Fort William the night before you can spend a relaxing day wandering around, perhaps going on a boat trip on to the Loch. As none of us had been to Fort William this seemed like a nice idea rather than seeing the place from a car window as we zipped through. Some groups doing a late start drive up on the day, although we think this puts a huge strain on your drivers as they would be driving for several hours to get there and then a further 11 hours over the next 24 hours.

The timigs and schedules above are all set for a Saturday/Sunday attempt which is when typically most people do the challenge. If you are planning to do the challenge mid-week you would also need to inlcude the traffic factor as it is likely you may get caught up in rush hour traffic somewhere unless you are very careful with your timing.