There's an old saying which goes "When is the best time to go fishing? Whenever you can!"
I don't know about season times in Scotland, but around my neck of the woods it always seems best in May and June. Even if you catch nothing it's beautiful to be down by the water at that time of year anyway.
If your father-in-law is a total novice to fly fishing I would recommend getting an hour or twos flycasting tuition organised proir to booking a full days fishing somewhere. Casting the fly is a totally different thing to casting ordinary coarse or sea tackle and even a normally competant angler can get in a frustrating mess if not used to the method. Most commercial trout fisheries have a tutor who's services can be booked by the hour. This might seem extravagant, but even a good fly fisherman can often benefit from a little coaching, and a novice will be able to pick up points that will hopefully lead to a less frustrating days fishing. Don't let this put you off the idea though. Casting a fly line is just a toally different action to other forms of fishing and requires a degree of timimg and an understanding of the method. This is where a little coaching works wonders.
One thing that I cannot stress enough for anyone fly fishing is that they WEAR GLASSES!
If the chap doesn't normally wear specs, get some polaroid sunglasses and wear those, but in any case, glasses are a MUST. Flies have a habit of being pulled around on the slightest breeze and can end up anywhere (especially with a novice caster). Getting one stuck in the back of your neck or in your ear is bad enough (don't ask me how I know!
) but one in the eye would most certainly not be good.