Thanks both of you. Around here peer supporters only help(ed) at support groups, so that was the situation I was thinking of; I of course agree I couldn't take them along to hospital with me. The group is in a children's centre, but this isn't coming from them, but rather the organisation now responsible for running the group.
Data protection - as I understand it's the fact there's a clipboard where the details of who attends are recorded. I don't generally even touch the clipboard. If I did, I know I shouldn't steal people's personal information 
Manual handling - yep, moving tables and chairs apparently. Perhaps the box of leaflets? And I could only speculate on the infection control stuff. We did have supervision sessions before and I think they are continuing, although I guess we'll no longer be able to bring our children along to them either.
My training was run by an NCT BFC, commissioned by the team running the groups. I have been eying up support groups in other nearby areas to see if there is somewhere else I could volunteer, although I haven't found anything at a time I could do yet.
The organisation now running the support groups is a Community Interest Company providing services for the NHS. I don't think they're setting out to make it impossible for us; they appear to have decided they need to treat us as unpaid employees, rather than treating volunteers differently. So if employees can't bring their children to work, neither can we, as their employer liability insurance doesn't cover employees to do so. If employees need to do this, that and the other training, then so do we. We had to fill in job application forms and have an interview - at mine I think they conceded that they knew other areas did not have these restrictions, but said they weren't big enough to get around them - not sure if they mean the groups themselves are too small, or the CIC.
Sorry for the slightly epic post. I am quite frustrated about it all so am ranting somewhat.