Fascinating! And thank you for ordering the pair.
I reckon that part of the confusion has arisen because Aston by Budworth is very close to the edge of the Altrincham Reg District, where it butts onto the Runcorn District. So nobody's been quite sure who should be reporting to who...
Joseph Donald was actually present at the death - he lives at Aston by Budworth, where the death is supposed to be, and he's gone to Samuel Carter, the registrar of deaths(?) for the Lymm sub-district of the Altrincham district. Notice that there's no certification here.
Thomas Gough of Budworth was "in attendance" on her and he's gone to Peter Edgeley, registrar of deaths(?) for the Budworth sub-district of the Runcorn district. This one is certified so I wonder if Thomas Gough knew the doctor? (The village of Great Budworth actually is over the border of the district and so is in Runcorn RD. He's told the registrar that the death was at Crowley, a township in Great Budworth parish in Runcorn RD).
Joseph Donald has gone quite a bit later on 11 January 1847 while Thomas Gough went on 4 January. Given that the death occurred on 27 Dec 1846, I would take a guess that Joseph Donald wasn't doing anything about it until something came up in conversation and someone told him that he should talk to the registrar - or the registrar heard of it and went to him. Never quite sure about how it's likely to happen. Since the 2 registrars are different, neither would realise that the other was also working on the case. We'll never know the exact circumstances but "muddle" seems a good word!
NB - my 1857 Directory has a Thomas Gough in Great Budworth as a boot & shoe maker. And Mr Joseph Donald at Aston by Budworth, a "private resident" at Birchbrook House. Which is kind of the opposite way round to what I guessed!