The point made by others, that soldiers were required to obtain
permission to marry was to get the wife and children on the 'regimental
strength' so they would be allowed to travel to a foreign posting. This
did not actually mean very much as if the husband died, whether through
injury or illness, the family were then left without means and one often
finds a widow re-marrying very quickly. It was their only way of surviving.
If you contact the Royal Engineers museum at Gillingham, Kent they may
be able to tell you which unit was stationed in Bermuda in 1880. I don't
know about the posting practices within this Corps but if soldiers
tended to stay with one unit, they may be able to tell you where they
moved onto from there.
Soldiers' documents for the Royal Engineers from 1873-1882 are held by
the National Archives at WO 97/1849-1857 so you may find something here
on line. Were any f the children born abroad? There are ways of finding
this information, if you have not found their births in UK.