Pendulum attached to an accelerating train The question I have is similar to that of the Pendulum in an accelerating train problem. Where a bob is hung from the ceiling of a train that is at rest. The train then begins moving with an acceleration "a".
I understand that the mass of the pendulum bob does not affect the period of the swing. This fact can be verified using the formula to determine the period of a pendulum undergoing simple harmonic motion.
Will there be time delay between when the train first begins to accelerate and when the pendulum bob first begins to move, using a heavier bob compared to a lighter bob?
 A: If you are asking how quickly the mass will respond to the train's acceleration, then the answer is instantaneously and this is independent of the mass of the bob. This is due to inertia and inertia is a property of all masses.
The period of a pendulum if it's in a stationary frame of reference is given by $$T = 2\pi \sqrt{\frac{l}{g}}$$ and the mass does not affect $T$ as you stated.
If you want to know if there is a difference in the value for $T$ depending on $m$ if the train accelerates, you must first ask if the motion of the train causes a dependence of $T$ on $m$:
Procedure:
If the trains acceleration is $a$, then the new acceleration of the bob, or its effective acceleration will be $$a' = \sqrt{g^2+a^2}$$ and you can get this by drawing vectors for $g$ and $a$ and use Pythagoras's theorem. This means that the new period will be
$$T = 2\pi \sqrt{\frac{l}{a'}} = 2\pi \sqrt{\frac{l}{(a^2+g^2)^{\frac{1}{2}}}}$$
Again, even though there is an acceleration, the period of the pendulum is still independent of the mass of the bob. So the answer to your question must be no. There is no difference.
A: In the non-inertial reference frame that is the accelerating train, the bob experiences a fictitious force $-ma$ where $m$ is the mass of the bob and $a$ the acceleration of the train.  This force is proportional to the mass of the bob.  Assuming the bob is initially hanging vertically, as soon as the train accelerates, the initial displacement of the bob in the $-a$ direction is instantaneous and is the same regardless of the mass, since the force is constant and proportional to the mass.  (This is similar to two objects of different mass dropped from a height; the objects travel the same distance in the same time- neglecting air resistance- since the force of gravity depends on the mass.)
For the pendulum motion in the accelerating frame, see the answer by @joseph h and also Time period of a simple pendulum in an accelerated frame on this exchange.
The following was added based on comments.
Assume the bob hangs vertically before the train starts to accelerate. When the train starts to accelerate there is instantaneous relative movement of the bob relative to the train seen by observers both on the train and on the ground. To the observer on  the train, the train is not moving and the bob moves due to the fictitious force. To the observer on the ground, the bob is stationary and the train moves.
In both cases the relative motion of the bob relative to the train is the same and the bob continues to move relative to the train until a component of the tension force (a) counters the fictitious force for the observer on the train and, equivalently, (b) equals the acceleration of the train for the observer on the ground.  The bob moves until it reaches a new equilibrium position. If the bob is "tapped" (slightly displaced from equilibrium), it acts as a pendulum about the equilibrium position. See Time period of a simple pendulum in an accelerated frame.
A: Previous answers are wrong in explaining when the bob starts to move... It is not instantaneous.
The bob is hung from the ceiling and it is the ceiling that is accelerating. The top portion of the string accelerates along with the train... But the portions of string down the line will start to displace only when that disturbance reaches them. More like when we create a wave on a string, the farther end displaces only when the wave reaches there.
Thus, the disturbance produced at the top end of the string takes some time to propagate down, characterised by the speed of transverse waves in that medium.
Now the speed of transverse wave in a medium is dependent on the tension in the string which in turn increases when you keep a heavier bob.
$$v=\sqrt{\frac{T}{\mu} } $$ where $\mu$ is the mass per unit length of string.
So if mass of the bob is increased, the tension in string increases, the waves travel faster and bob starts to move quicker
