The nominal groove diameter of the .375 Winchester is usually 0.375"-.376".
Bullet and groove diameters of .38-55 rifles can vary considerably from the nominal 0.379"-.382".
Assuming that your reloaded .38-55 ammo will chamber easily, they willl shoot just fine in the .375 Win chamber. The larger diameter cast lead bullets may not chamber easily.
I find that using .30-30 cases blown out to .375 Winchester and loaded with 0.377" 250 grain RNF lead bullets gives the best accuracy. Powder charge will be that of of a .38-55, of course. Avoid hot .375 Win lead bullet loads. Jacketed .375" bullets will give satisfactory hunting accuracy, but nothing spectacular.
HTH
John