The appropriate and fastest library depends on several things. Which Bessel functions (only J, Y & Hankel or modified Bessel functions I & K too), for which types of arguments (real or complex, integer, fractional or general order)?
Amos's libraries are written in Fortran-77 (there are Fortran-90 coverted versions of TOMS 644 on a mirror of Alan Miller's fortran page: http://jblevins.org/mirror/amiller/) and are numerically accurate and fast for complex argument of all Bessel functions, but they are difficult to understand the source code. He produced several versions of his library, but only the one included in SLATEC (http://www.netlib.org/slatec/) is actually open source, the other versions have a copyright associated with the transactions on mathematical software journal.
mpmath (http://mpmath.org/) uses a totally different and more general approach, by expressing all Bessel functions as special cases of hypergeometric functions. Mpmath is also arbitrary precision, so this is probably overkill if you just need double precision. This is more general, but slower. The author of mpmath has a C-based library called arb (http://fredrikj.net/arb/) that implements some of the functions in mpmath but much more quickly (since it is C-based rather than Python-based). Since arb and mpmath are arbitrary precision, they might be good for benchmarking, but probably not for speed testing.
I haven't personally worked much with the Gnu Scientific Library, but I believe it is fast and accurate for what functions it covers. It may not have all the functions you need. It would probably be a better example of well-written modern code, compared to the Amos libraries.