You are in the right direction, as the AppleTV box is one of the best solutions out there.
I believe what you'd like is not only for all your TV to be on one platform.. but also under one app. Well the bad news is that this can't be done right now...
However, Apple are providing the next best thing by way of its 'TV' app.
All you need to do is use the TV app to search for shows - and currently the TV app will work for BBC, ITV, C4, C5, Amazon Prime, Britbox, UKTV player, and Disney+. Any shows you add via the TV app will be consolidated into its own 'up next' list and it works very well.
Netflix also works in but in search only.... and you can add a Netflix show to the TV app, however it will not work with keeping track of viewing progress and provide the next episode - it is at best a dumb link to that show but I suppose better than nothing.
Unfortunately Netflix, and NowTV dont play nice with the TV app - but we were promised that NowTV would... so if that promise is to be kept, then the TV app is the answer to all your problems.
Until they do keep their promise... the TV app is great for everything apart from NowTV so you will find yourself having to visit the NowTV app explicitly for its content.