Sika can be very wary if they are shot at. I've no experience of reds in lowland/woodland situations and very little experience of them on the hill so I really can't make any valid comparisons.
Where I shoot them the last half hour of legal shooting time is the only time you'll see them out and about in most places, especially places where you might get a shot at them. However, hinds and small stags can be nosy and this can be their downfall as they will often try to work out what it is that disturbed them. Big stags just run, they don't look back or any of that stuff. However, if they can avoid it none of them will make a habit of coming onto open ground but rather will try to feed in areas within the wood or along the edge so they might only rarely come onto the fields if there is feeding in the wood.
I also believe that they have very good eyesight, they will see you standing still in cover at 300 yards with annoying regularity and I believe one big stag spotted me at 750 yards - a friend and myself were watching him with a spotting scope when he looked straight at us (the wind was good for us) and ran back into the trees. Make of that what you will but we were convinced that he saw us and we were lying in the heather.
I've also found that they like certain places, and dislike others, for no clear reason and they can change their mind and just vanish, they will move considerable distances overnight if they decide to go. Also at some times of they year, seemingly randomly, they can just move to somewhere else and stay for several months. This doesn't help if you are trying to work out a pattern so just because you sit and watch that forest edge for a week and see nothing doesn't mean they won't be there the following week, or every day in July or something equally random.
In saying all of this where there is less shooting/poaching pressure than where I shoot them I've seen them out in fields in broad daylight. Recently I was on the south side of Loch Ness and was completely stunned to see them stand around in fields in daylight and not run if you stopped the car for a look. So, I think the level of shooting pressure is a very significant driver of their behaviour.
A positive note is that sika seem to be able to thrive even under significant shooting/poaching pressure. Here in Ireland roe wouldn't last 15 minutes if they were to be introduced again and reds just about manage to hang on in many areas whereas sika do well and are expanding their range and numbers. So even if the FC think they are hitting them hard it is likely the population is increasing and that the deer will move out to other areas. They will be very careful deer but in your position I'd be hopeful that they are around on your ground for at least short periods each year and, more likely, they are there all the time but the numbers are low and you just haven't seen one yet.