More4+1 is transmitted by the PSB3 'multiplex' (= 'bundle of channels'). PSB3 is an HD multiplex, so channels transmitted within it are only available on those TV sets which have an HD tuner built into them. (That applies even if the channels themselves, such as More4+1, aren't actually transmitted in HD).
A TV set which carries this logo on it (or on its packaging) should be able to handle channels transmitted via an HD multiplex:
https://tinyurl.com/y4vt6hke
The same applies if you see this logo instead:
https://tinyurl.com/y4shm8ll
However if you only see either of these logos
https://tinyurl.com/y4nlhhrc
then the set hasn't got an HD tuner built into it and won't be able to receive channels transmitted via an HD multiplex. (That applies even if the set is described as 'HD ready'. That simply means that the set can display HD signals from an external device, such as a Blue-ray player. It's not the same as there actually being an HD tuner within the set itself).
The contents of your post suggest to me that your 'ageing Sony Bravia' is a true HD set (and not just 'HD ready'), whereas the other two TV sets aren't. If I'm right, you should find that Channels 101 and 102 on the first set display BBC One HD and BBC Two HD respectively, whereas they're blank on the other two sets.