I think we will all have to have POA, but it doesn't seem fair when I am doing everything. There's no benefit to being named as an attorney on a PoA, so no unfairness about siblings being named but not doing anything. All it does is give them a responsibility over matters which they've chosen to have no control - not an enviable situation.
As others have said, it's up to your Mum, and her alone, to appoint attorneys. Sister can't insist on anything. But it might be useful to have them as "spares" in case anything happens while you are on holiday or otherwise unavailable - if you're doing most of it, PoA needs to be set up so you can act "jointly and severally".
There's two stages to the PoA - first stage, once it's drawn up and signed, is to get it registered with the Court of Protection - useful to do this immediately, not wait for when it needs to be used.
Second stage is to allow any organisation you deal with to see a copy of the registered PoA and to register you as an attorney on their records. They don't have to register all the attorneys in one go (or at all). My DF has both me and my DH as attorneys, but it's only me that deals with his finances, and so we've not bothered to get DH registered with the bank/building society. If I went under a bus, DH could take the PoA in to the branch along with the usual id and get himself registered and take over DF's affairs, but meanwhile it's simpler - it means I can open a new savings account for DF without having to drag DH along.
You can't have two PoAs operating at one (imagine the confusion!) so if cousin does have PoA, then that needs sorting out. It looks like you would need to revoke the first one using a Deed of Revocation. www.willpack.co.uk/revoking-an-epa-or-lpa/