Five years and no fruit in all that time ?!
I would agree it most looks like some sort of cherry. Potentially a non-self-pollinating (or an ornamental) variety!
You haven't said how big it is william_wt, but there is always the option of a drastic prune, to allow the fence repairs, rather than removal. Drastic pruning of a mature fruit tree (or any tree) cannot be recommended - bit by bit size-reduction is better done over a number of years - but yours is not fruiting and obviously a hindrance to your fencing repairs so perhaps worth the risk of a substantial prune to suit your needs.