We cannot talk of an object to be "not accelerating". We can only talk of an object to be "not accelerating with respect to a specific reference system in which we measure position and time". One often avoids such long sentences to make life easier. However such attempts to make life easier may lead to misunderstandings.
Example: You are sitting in a sports car and push the gas pedal. The sports car accelerates with respect to the reference system of the road. Also you accelerate with respect to the reference system of the road. However you do not accelerate with respect to the driver's seat. That is why you stay in the drivers seat (you are not accelerated wrt the driver's seat).
In order to describe the apple example correctly, we thus have to choose a reference system first. The description we will make will depend on the reference system and on the physical theory we use.
Let's use the point of view taken by general relativity, since it is a good theory for understanding gravitation. Here, gravitation is not considered a force but part of the surrounding geometry.
Now let us choose the reference system of an apple in free fall. This falling apple feels no force and it is not accelerating WRT to the chosen reference system. The falling apple will look up to its brothers and sisters which are still attached to the stem. The apple will say: "My brothers and sisters are pushed upward by the stem and it is due to this force that they are accelerating upward wrt to myself". In GR a free falling reference system is an inertial system. So the observation also is consistent with the statement that "force leads to acceleration" only applies in inertial systems.
Now let us choose the reference system of the apples still attached to the tree. These apples feel an upward pulling force from the stem. They are astonished that they are not accelerating wrt to the chosen reference system. However they understand that the law "force leads to acceleration" only applies in inertial systems. So they see no inconsistency, since they are not in an inertial system.
Let's now use the point of view taken by special relativity. Here, the notion of an inertial system is different and that of gravitation as well. Gravitation is now a force and a freely falling reference system no longer is an inertial system. The explanation still is consistent inside of the theory.
First let us choose the reference system of the apple in free fall. The apple is acted on by the gravitational force. However the apple is not accelerating. This is fine. In SR a freely falling system is not an inertial system and so we do not expect "no acceleration = no force". The apple is feeling no force. This is fine as well, since the gravitational force is compensated by the inertial force which is due to observations being made in an non-inertial system.
Now let us choose the reference system of the apples still attached to the tree. This is no inertial system either (rotation of earth around its axis, rotation of earth around the sun). However, for our purposes we can neglect these effects, since they are small. We have an inertial system here, more or less. The apples sitting in the tree will argue: We are acted on by gravity and this is compensated by the forces of the stem. So we are not acted upon by a net force. We feel the tension between the gravity and the stem in our necks, however, but net force is zero. Since we are in an inertial system, the concept applies that no net force means no acceleration wrt this system. This is fine. The apple falling down is pulled by gravity and is accelerating wrt to this inertial system. This is fine as well.
The confusing thing is: It depends 1) on the chosen reference system and 2) on the chosen theory, how we have to reason. The arguments always look different and the concepts as well. For example, in general relativity, gravity is not considered a force. Still, in the chosen framework we can argue consistently.
If we are familiar with the concept of a geodesic in space-time, we can even extend the reasoning to this setting. The freely falling apple is moving on a geodesic and feels no force. The apples on the stem are pulled by the stem from the geodesic and feel a force.
Edit: Since you edited the question - let me edit the answer as well. :-)
An accelerometer attached to the apple on the tree shows a readout of 1g. An accelerometer attached to the falling apple shows a readout of 0g (neglecting air flow, assuming vacuum, neglecting field inhomogeneity). Fun experiment: Go for a skydive (and you will experience this yourself).
Staying in GR and using geodesic: We can construct a generic reference system when speaking geodesics. This is locally inertial in the GR sense. The free falling apple is unaccelerated wrt to this geodesic reference system (and stays on the geodesic), it feels no force. The stem apple is accelerated wrt to this geodesic reference system (and therefore deviates from the geodesic in its movement), it feels the pull of the stem (and in this model no gravity, since this is assumed to be in the geometry).