I don't personally know of any accounts of anyone taking the tube to a hanging, but I can certainly believe it would happen. One of the oldest tube stations, the Barbican Underground Station (known then as Aldersgate Street, opened 1865), is a mere ten minute or so walk away from where Newgate Prison was once situated, and it would be outside the prison that people were hung, the same time it would take to walk from Farringdon Station, opened 1863. It would be a 15-20 minute walk from Cannon Street Station (opened 1866), passing Parliament, or a 20 minute walk from Moorgate (opened 1865).

All in all, it would be practical for at least some people to have taken the tube, particularly to Aldersgate Street or Farringdon Station, to view the last few public executions at Newgate. Another popular site of execution, Horsemonger Lane Gaol, was a minimum of 35 minutes walk without much pedestrian traffic, and on an execution day, even in the mid 19th century, despite there only being 130 or so executions in the entire 1860s, public executions were popular and the crowds stretched for miles, so walking would take hours more than the modern day walk from the station.

So this is very short as we don't have any accounts, but it's likely that people did take the tube to see public executions at Newgate, unlikely for them to take the tube to see public executions at Horsemonger Lane and impossible for other public executions full stop.