I have two trees in the yard. On in the front and one in the back, and
there is a real problem with surface roots, that cause a virtual
ponding in the front and the back yard. The tree in the front yard is a
japanese cherry blossom tree, and a root grows across the yard, where
the water is prevented from flowing away from the house, towards the
street. The tree in the back is a tremendous oak tree, and there is a
huge root breaking the surface, and prevents water from flowing down
the gradient into the neighbors yard, which is lower than mine. You can
see where the water collects.. and you can actually feel that area is
lower than the rest of the yard.

Do I have to have those trees cut down right off the bat to alleviate
this problem? What would happen if I just cut away those problem roots?

Wow, I have to say the answer is no. Those sound like some nice mature
trees that have to add a lot of value to your property, not to mention
the pleasure they must give you. If you cut away the roots of a tree
my guess is that it would kill them. You should consult a certified
arborist to determine what your choices are for managing those valuable
trees.

You say that your problem has been largly solved except for major
rainfall events. You could just install equipment in the basement
like a sump pump or de-humidifiers as an alternative to destroying
those trees.