No manufacturer or retailer is allowed to sell anything marketed as a TV without it having a Freeview tuner. It's mandatory. This has been a requirement since 2008. There's no "99%" about it. If there's no Freeview tuner then it's not a TV, it's a monitor.
At 40" and above, all the TVs will have Freeview HD tuners and have IPTV features, so they'll be Freeview+ sets where they integrate catch-up TV as well as over-the-air TV reception. Question: Does your mum have an internet connection? Does she pay for broadband?
The reason for the question is simple. For UK TV services - the stuff that comes via an aerial, but it can include a satellite TV version called Freesat - nothing broadcast live is better than 1080i resolution. That's the HD standard that Freeview and Freesat use, and there aren't that many HD channels. Most of what she'll watch will be standard definition. 576i resolution. If the plan to change the telly is to get streaming and catch-up services, then that's fine. A new TV will do that. But if the idea is that a 4K TV will look better than the old set for everything, then whoa there horsey. That won't happen.
"Wah?? A 4K TV can look worse?" - Potentially, yes.
Here are the most common display resolutions since digital TV became a thing. We have Standard Def (SD) at 720 x 576 pixels. HD Ready LCD/LED TVs tend to be 1366 x 768 res. Then you have Full HD at 1920 x 1080. Top of the shop for most folks is UHD (4K) at 3840 x 2160 pixels.
There's still a heck of a lot of TV programming that was made at SD resolution. Any time that stuff is shown on a TV with a higher screen resolution, the TV processing has to upscale the image to fill the available pixels. There's a big issue with this. These old programmes were never intended to be upscaled. The pictures don't stand up well. The higher the resolution of the TV screen then the harder it is for the TV to upscale the image. A UHD TV is 20 times the resolution of SD. That means that when an SD image is upscaled to UHD, then between two adjacent SD pixels there's a gap of 19 empty pixels where the TV has to best-guess what should fill that space. This is happening 25 times a second for about 7.9 million pixels each time. That's a lot of maths.
The quality of the scaling is important. The TV settings can also play a part too. An example of this was the F1 qualifying I was watching a few days ago on the canteen TV at a site I'm working at. I think the screen was an LG 43" or 50". Every now and then a list would pop up with the driver's lap times. The text was barely readable. It didn't get much better up close. I had a fiddle with the settings; turned the sharpness down and switched off a lot of the picture 'improvement' processing. It made a big difference to fine detail. The text became a lot more readable.
The rugby was another casualty of poor picture processing. From a distance it was okay, but from 10ft away it was easy to see all the MPEG compression around players.
A new TV will possibly be brighter if you avoid the base models from LG and Samsung. If your budget ceiling is £500 then have a look at the Sony KD43X75WL. After that there's a bit of a jump in price to around the £800 mark.
Where your mum doesn't have an internet connection, (or won't have Virgin is ditched) then think twice about changing the telly. Where the old one is still going strong then a new budget set might be a step backwards.
Regarding access to signals, she'll need either a satellite cable (or two if recording is required) plus a Sat receiver box which itself will be £230. Freeview reception just needs a signal into the TV from an aerial feed. If access and permissions are not a problem then a local installer should be able to rig up a roof aerial plus downlead for around £160. This will be all that's needed if she decides to add Freeview recording (old Humax box?).