To be able to re-marry, the deserted person would have to have their spouse legally presumed dead:
"England and Wales[edit]
Currently,[when?] the law generally assumes a person is dead if, after seven years:
There has been no evidence that they still live.
The people most likely to have heard from them have had no contact.
Inquiries made of that person have had no success.[2]
This is a rebuttable presumption at common law—if the person subsequently appears, the law no longer considers them dead.
Otherwise, courts may grant leave to applicants to swear that a person is dead (within or after the seven year period).[2] For example an executor may make such an application so they can be granted probate for the will. This kind of application would only be made sooner than seven years where death is probable, but not definitive (such as an unrecovered plane crash at sea), following an inquest (see below). Such an application is specific to the court where it is made—thus separate applications must be made at a coroner's inquest, for proceedings under the Matrimonial Causes and Civil Partnership Acts (for remarriage), for Probate, and under the Social Security Act."
from:http://en.wikipedia.org/wiki/Declared_death_in_absentia
I presume on their spouse being presumed dead, they would then be a widow/widower and so that description would be what is entered on any subsequent marriage certificate
