Getting food into our LO's is definitely a stressful thing sometimes! When you say he is sicky, or that he is sick at meals, what is making him sick? Does he have something specific like reflux? Have you addressed it with his doctor? I'm just wondering if there is more going on. I noticed on your routine he only has 2 BFs a day. Is he getting enough milk do you think? Just want to make sure he's still getting enough nutrition.
Definitely try to take a relaxed approach to mealtimes as you don't want him to develop any negative associations with eating. For the holding food in his mouth, DS1 has gone through phases like that as well and they have always passed, so hopefully it will pas for you as well.
The fact that he's eating something at nursery is good, and it means he still wants to eat so he might just need a change in the eating environment at home to get him interested in mealtimes again. Do you eat your meals with him? It might be a good idea to try as much as you can to eat together as a family, or even just you and him. Even if it's not your mealtime, sit with him and have a snack, and try to make it an interactive and social event, rather than you sitting there feeding and watching him, kwim? If you haven't already, start letting him eat the same meals as you do, just without any added salt or strong seasonings. He might be more interested in eating 'big people's' food.
Often my DS won't want what's on his plate but will want the exact same thing off of ours!
If at some point he is really not interested or is really fighting eating anything, I would end mealtime and get him down rather than dragging it out longer, that way hopefully neither of you gets too frustrated.
Of course you will want to keep encouraging him to eat, and offer a good variety of tastes and textures. Try to make it fun. But try not to stress as well. Make sure he is still getting enough milk (but not too much as you still want him to be hungry for food, just the recommended amount for his age). You are entering toddler territory and toddlers surprisingly often get away with eating very little, much to mummy's dismay!