Hmm, depending on what you mean by 'only nutritious' processed foods and exactly how strict you would plan to be, I would say it is possible to feed a family this way, but would it really be worth it? Why would you be doing it? What health benefits are you hoping to get out of this and can they be achieved in another way?
I really do think that processed foods have become demonised on MN and in the media to an unfair extent, yes eating loads of shop bought cake filled with lard and sugar will make you fat and give you diabetes, heart disease and other nasties, but eating loads of lovingly homemade 'wellness balls' (or some other suchlike recipe peddled as 'healthy' by Ella and her ilk) made with coconut oil and date molasses will do exactly the same. Honey and fruit derived sugar substitutes are still sugar at the end of the day, yes delicious but not exactly health foods. Sadly very few things which are delicious and moreish like cake and biscuits are healthy, this seems to be sod's law (well maybe some lucky souls genuinely relish and gorge themselves on carrot sticks and celery but I have yet to meet that person!). So would you just be substituting foods you enjoy and are easy/work for the family, for 'unprocessed' versions which are harder to make, less tasty and not much more healthy?
If you really want to improve your/the family's diet with a focus on health and feeling well, there are other changes I would look at making first rather than worrying about processed vs non processed foods. So rather than denying everyone the treats you all enjoy, could you think about having them at the weekend only, or if that's too much, every other day to start with? On non treat days have fruit or natural (unsweetened) yogurt as a pudding?
Swap sugary/fizzy drinks for water, or again reduce down and swap for diet versions or limit to the weekends only at least as a starting point?
Reduce alcohol to within NHS guidelines, have at least 2 or 3 alcohol free days a week, or ideally stick to alcohol only at the weekend?
Reduce red meat consumption and replace with more fish/chicken/pulses as protein options?
Increase veg consumption so you are consistently getting your 5 a day? Base meals mainly around veg with moderate portions of lean protein and complex carbs - ideally wholegrain?
Increase exercise levels and regularly do a physical activity (walking, biking, swimming?) as a family?
Introduce some wellbeing/mental health improvement practices, e.g. mindfulness, meditation, daily gratitudes, journalling, self-care, positive thinking?
Apologies if you already do all of the above but if so then I would say you have a pretty healthy lifestyle already and don't need to sweat a few treats on holiday. Or on the other hand if all the above seems overwhelming/really difficult then remember that even making one of those changes, most of the time, would bring definite benefits to the family (more so than faddy or impossible to maintain 'clean eating' anyway) - so focus on one thing at a time and small, achievable steps rather than trying to achieve perfection in one step....