Replacing a distributed sky setup with just Freesat?

slightly off topic but most of the rte player tv apps dont let you stream live fyi

Yes, you can just swap out a Sky HD+ box for any FreeSat box. (If you have Sky Q only certain boxes will work unless you change the LB).

[I assume when you say the output is distributed to 3 rooms, you mean there are 3 feeds from the Sat dish to each room]

There are a range of boxes available depending on what you need (recording, storage, apps etc.). Just go onto Amazon, check the reviews and pick something that suits your budget. Just don't expect to spend 50 Euro on a box and get as good an experience as Sky.

The satellite signal used by Sky will be exactly the same as that used by the FreeSat box. So if you have a good quality signal now, it will be the same with FreeSat. I wouldn't worry about upscaling.

Can you just clarify when you say "one box in the attic and distributed to three rooms" are you using RF from the one box so that it is tuned into each TV as a channel and all rooms can only see whatever the sky box is watching?

As I feared you are using the RF output of the Sky box to feed the other rooms. I am not aware of any contemporary FreeSat or FTA boxes that have RF out. Others may know of some. It is possible to get HDMI to RF converters but they are almost as dear as the new box. As an aside, by my understanding of your setup, you would never have had a HD picture on the remote TVs even if watching a HD channel. Again I'm open to correction on that.

Slightly different approach would be to invest in something like this,

I bought one nearly two years ago and works quite well, it is slowly getting less buggy with software releases.

It integrates with Plex Pass/Premium but as a Andriod box itself not great