How long after the spouse's death should one seriously contemplate on getting into a relationship with a widow(er)?

How can anyone answer this for someone else? 

There is no should.

Ideally just before the spouse's death. If you wait until after the death, everyone else will be after the widower too. 

...

Not sure why you would confine yourself only to widows but I think that is very personal. Some men who have become accustomed to being cared for by women I notice marry fairly quickly the first suitable person who will suitably care for them which sometimes is their hired cook or housekeeper.  Women are more  likely to wait a couple of years before getting into a new relationship.  And some people just seem to have no desire to seek another close relationship.  After my father died my mother did not. Perhaps they are very set in their way so unable to accommodate anyone new.  And your children would have input as well which perhaps they should not but they easily can sway a parent's decision.  They would have their own fears about your seeking someone else though they mostly of course want you to be happy. 

I would say best to examine your feelings as well as your possible feelings about someone else and about your desire to form a close bond with them.  I think people need people so  I would say less good to be alone.  On the other hand if we are older and so set in our ways that we are no longer able to alter our lives to accommodate someone new then that would be more difficult.  

You can "seriously contemplate" the relationship anytime you want.
All you're doing is THINKING about it.  

Contemplate has various meanings.  One is to view or consider with continued attention : meditate on. 

A second is to view as likely or probable or as an end or intention.

If one has just been recently married, I think that "during the honeymoon" would be premature.

As soon as she's drunk.