Since there's a thread linking to the NHS volunteers site I thought the Council scheme is also worth a plug: https://www.barnsleycvs.org.uk/ Sorry not been on BBS last few day so don't know if this has already been covered. There will likely be some overlap at this stage and I don't know the advice re: registering as a volunteer in multiple places, but I expect there to be a clearer division of responsibility in the coming days.
Not sure about Barnsley, but anyone wishing to volunteer in Doncaster needs to join what's shaping up to be a sensible, credible scheme through the Council. Looking after your own family and friends is great, but to help those you don't know, register with the Council...local Cllrs should be able to put you on the right track.
That's exactly what I've linked to above. I'd urge people to join - the facebook groups etc are great, but the council are better geared up to match volunteers to need at scale.