How does a photon move in an electromagnetic wave? Electromagnetic waves by definition has a changing electric and magnetic field.
Photons are sent radially outwards by a charge and thereby constitute the electric field.
Then how could an oscillating charge vary the field strength of each point in space?
 A: 
Electromagnetic waves by definition has a changing electric and magnetic field.

This is the classical framework, the electromagnetic waves are solutions of Maxwell's equations.

Photons

Photons are quantum mechanical entities and belong to the quantum mechanical framework.

are sent radially outwards by a charge and thereby constitute the electric field.

No. A static charge is not sending anything anyplace. Classically  it has an electric field  which describes the topology/map that another charged particle will find when approaching the first one. A single charge just has this mathematical description of an electric field.
You may be thinking of the approximation of large wavelength photons used as an illustration of static fields. Have a look in the answer by Motl here for this. (which uses the concept of virtual photons, i.e. mathematical descriptions of photons).

Then how could an oscillating charge vary the field strength of each point in space?

Still in the classical frame, the field changes as the charge moves, i.e. the energy in the motion of the charge creates the changing electric field and because it is a moving charge, a changing magnetic field, which independently constitute the electromagnetic wave.
At the quantum level, the electromagnetic wave is built up by a confluence of a large number of photons, as explained here. . It is not simple and depends on the boundary conditions of the problem under study. This answer might help in understanding.
For your particular example of a moving charge, a single photon carries away energy and momentum from the energy and momentum of the motion of the charge, as it accelerates/decelerates with the oscillating motion. Due to the synchronicity of oscillation the phases of the  individually created photons are such as to build up the classical electromagnetic wave.
