The results of the mathematical experiment done with Maple (The DirectSearch package should be downloaded and installed in your Maple.)

f := proc (N) options operator, arrow; DirectSearch:-Search(proc (x) options operator, arrow; min(seq(BesselJ(n, x), n = 1 .. N)) end proc, {0 < x}, maximize) end proc:
 seq(evalf(-ln(f(N)[1])/(ln(N)*N)), N = [2, 10, 20, 40, 100, 1000, 10^4, 5*10^4]);

$$.5564145611, .3883486929, .4924980215, .5720147732, .6487478567, .7617480189, .8208958156, .8474981352 $$ suggest the dependence $$\log(f(N))\approx \frac {-1} {N*\log(N)},\,N\to\infty. $$

The results of the mathematical experiment done with Maple (The DirectSearch package should be downloaded and installed in your Maple.)

f := proc (N) options operator, arrow; DirectSearch:-Search(proc (x) options operator, arrow; min(seq(BesselJ(n, x), n = 1 .. N)) end proc, {0 < x}, maximize) end proc:
 seq(evalf(-ln(f(N)[1])/(ln(N)*N)), N = [2, 10, 20, 40, 100, 1000, 10^4, 5*10^4]);

$$.5564145611, .3883486929, .4924980215, .5720147732, .6487478567, .7617480189, .8208958156, .8474981352 $$ suggest the dependence $$\log(f(N))\asymp -N\log(N),\,N\to\infty. $$