If this is the same man (which is far from certain), I think it would be unwise to assume he was removed under the poor law from Westminster directly to Nantwich. There is a four-year gap between the dates, and he could have been anywhere in the meantime. He may have been in the Fulham Road infirmary only very briefly, and then moved out of the area. He is unlikely to have been in the care of Westminster all that time, because you haven't found another workhouse admission for him in the online records.
I wouldn't be too worried about the age discrepancy in the death index, since his age at death would have been taken from the Nantwich workhouse/infirmary records, which are frequently inaccurate. If you want to chance your hand, you could order the death certificate in the hope that there is something on it (e.g. an occupation) that helps identify him. But you should be aware that there may be nothing conclusive to show for the money spent.
