Toe-in is slightly different on radials.

... and increased positive camber.

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There is something to this, mounting radial tires on the original wheels that were designed to be used with tubeless bias-ply tires. However, a member of my Tbird club drove his 57 Tbird from Sacramento to Washington and back on radial tires on the original 57 wheels without issues. What can happen is that the "flex" can cause full wheel covers to come off. 

I had a problem balancing Diamondbacks and original Ford 15" Tbird rims.   When I put the new tires on the shop told me they had a problem with two tires and they installed 5 weights on each tire.  I brought the two wheels/tires to another shop that fixes modern wheels.  They couldn't do much better.  They told me the tires AND rims were out a little.

There is a balancing machine called the Hunter Road Force.  This machine puts pressure on the tires while they are spinning and measuring the balance.  The machine can tell the tech how to "re-orient" the tires on the rims so the imperfections counter balance.  After they did this the balance was much better.  Went from 5 weights to 3.  Not many places have this machine.   I went to a Ford dealer and they charged me 1/2 hour labor each @$90.   My only other option was to locate "better" rims......   Old perfect rims are hard to find and the prices seem to be $75 and up......   

https://www.hunter.com/wheel-balancers/road-force-elite#overview