What is the formula for the bank angle required for a turn in line-abreast formation? Imagine we have a line-abreast formation of aircraft, with differing airspeeds, turning around a central pivot point, like points on the radius of a phonograph record, with an equal amount time per 360 degrees of turn for each aircraft.  What is the formula for the bank angle required of each aircraft, in terms of the variables 1) turn radius and 2) time period per 360 degrees of turn?

Note that essentially the same situation applies when a slow-flying glider-- maybe even a hang glider-- and a faster, heavier glider are thermalling together and trying to stay exactly opposite each other.  Again the turn rate must be the same for each, but the turn radius is not, and the required bank angle for each might be different.  Clearly, the difference in turn radius is much more significant in this case, than in the case of high-speed aircraft turning in tight formation.
Turning Rate

All planes flying a turn in line-abreast formation are turning at the same rate $\omega$.

A standard turn is commonly defined as $\omega_1 = 3°\frac{1}{\mathrm{s}}$. 

Full Circle Time Period

If planes are turning at the same rate, the time period to complete a complete circle will be the same as well.

Time period: $$ T(\omega) = \frac{360°}{\omega}$$

The time period to complete a circle at this standard turn rate is $T_1 = \frac{360°}{3°}\mathrm{s} = 120\,\mathrm{s}$.

Air Speed

Since the outer planes have to travel a longer distance to complete their larger circle in the same perios, the outer planes have to fly at a higher airspeed then the inner planes. 

Air speed: $$ v = \omega r$$

Bank angle

Turn rate: $$\omega = \frac{v}{r}$$

Banking angle:  $$\tan{\theta} = \frac{v^2}{rg} = \frac{\omega^2 r}{g}$$

Gravitational constant: $$ g = 9.81 \frac{\mathrm{m}}{\mathrm{s^2}}$$

That gives the banking angle as a function of turning radius: $$\theta(r) = \arctan{\frac{\omega^2 r}{g}}$$

See the purple curve in the graph below for the bank angle in a standard rate turn (3°/s):


This curve seems to imply that one can fly a standard turn (or any other given turn rate) at any radius from 0 to infinity. Therefore I added two more curves to indicate the physical limits:


air speed (green) must be in the operating range of the plane
g load (blue) must not exceed the maximum acceptable for plane and occupants. (The correct scale for the g load is the purple scale on the left divided by 10)


And finally, the same plot but for a double rate turn (6°/s):


Practical Relevance

Typical turn radii are in the order of 1000s of meters. Typical distance (wing span) of planes in a tight formation is in the order of a few 10 meters, that is a few percent of the turn radius.

If you compare the banking angles for two planes flying in tight line-abreast formation, the banking angle difference will have a similar relative difference, that is a few percent, or a fraction of 1° in absolute terms.

[I will calculate and insert an example here, when I have the time.]

For the practical purpose of actual formation flying, pilots will give the arcus tangens a break, and fly seemingly identical banking angles.