You have 8 variables that appear in 12 combinations:

vars = {alpha1, alpha2, beta1, beta2, gamma1, gamma2, delta1, 
   delta2};
combs = {alpha1 - alpha2, beta1 - beta2, gamma1 - gamma2, 
   delta1 - delta2, alpha1 - beta1, gamma1 - delta1, alpha1 - beta2, 
   gamma1 - delta2, alpha2 - beta1, gamma2 - delta1, alpha2 - beta2, 
   gamma2 - delta2};

Actually, the combinations are not independent: all of them can be written in terms of the first six of them:

replaceWithC = Solve[combs[[;; 6]] == Array[c, 6], vars[[2 ;; 7]]][[1]]
combs /. replaceWithC

Your 8 equations can be rewritten in terms of 6 unknowns:

eqsWithC = N[eqs] /. replaceWithC // Chop // Rationalize

Your equations are either redundant or incompatible, and this may help explain why NSolve has trouble with them. The first four equations can be solved independently of the others:

sol4 = NSolve[eqsWithC[[;; 4]] && -2 Pi <= c[1] <= 2 Pi &&
    -2 Pi <= c[2] <= 2 Pi && -2 Pi <= c[3] <= 2 Pi &&
    -2 Pi <= c[4] <= 2 Pi] // Union

You can then solve the next two and see what happens with the last ones:

sol6 = eqsWithC[[5 ;; 6]] /. sol4[[1]] // Solve
eqsWithC[[7 ;;]] /. sol4[[3]] /. sol6