BBC HD looks dramatically different from BBC regular (SD). The difference is discernible on small and large HD TVs. The difference you’ll notice does depend on being close enough to the screen though. Opinions on optimum viewing distance vary but you want to be close enough not to see the individual pixels and not so far you can’t resolve fine detail in the image.
If your single strength is good enough (your freesat box should be able to tell you this) then things to look at include:
connection between FreeSat box and TV is by HDMI cable.
you have your TV set up to show the right HD format (automatic, 720P, or 1080P. Ignore 1080i it is awful)
your FreeSat box is outputting at the correct resolution (automatic, 720P, 1080P) e.g. some boxes default to 576P or 576i which are enhanced definition and standard definition respectively
You are looking at the correct channel (really. I have seen folks complain about pixellated images only to find out they were watching SD and expecting HD)
You have turned off all the dreadful ‘smoothing’ algorithms on the TV (sometimes called ‘sport mode’) these essentially blur the frames into each other.
The difference between SD and HD is like night and day. HD looks like smooth, crisp, moving photographs. SD looks like TV used to.
Any plans to move to 4K?
Have you tried comparing your FreeSat box against e.g. an HD laptop signal or other kind of set top box and e.g. Netflix?
Good luck.
[edit] as folks have pointed out, if a program was not shot in HD no amount of post-processing tricks will make it HD-quality. Gardener’s world is one program where the SD and HD contrast is marked. Especially on the close-ups and extreme close-ups of plants and insects.