Geochronology emcompasses many techniques (SIMS, TIMS, EPMA, LA-ICPMS etc) and many textbooks will focus on one, or a small selection. The techniques themselves may cover an awful lot of applications, major and accessory phases and rock types, and therefore an awful lot of labs and researchers.
I would suggest reading up a bit first (see above thread), and then looking at some papers on topics of particular interest to you. That way you may be able to see for yourself which labs are producing the high quality work in those areas, e.g. by looking at info such as publication citations, and what labs are used by the main people in your field
Gunter Faure or Clude J. Allegre : Isotope Geology, Dickins: For radigenic isotopes. These are the books which can be studied for the basic understanding of geochronolgy.
It's also worth looking at Uranium-series disequilibrium: Applications to Earth, Marine and Environmental Sciences, 2nd Edition, edited by M. Ivanovich and R. S. Harmon. Whilst being rather old, published in 1992, it's still covers the basics very well.
I also recommend the one mentioned by Larionov "Isotope Geochemistry"
by William M. White,
ISBN: 978-0-470-65670-9, December 2014 Wiley-Blackwell 496 Pages.
It is up-to-date, simple and good for beginners.
Second one is "Principles of Isotope Geology", by G. Faure, ISBN-13: 978-0471864127, 19986, Wiley, though a bit old.
For labs that do geochronology analysis, it depends on what you want to measure. So, first specify the method that applies to your samples and then you can find the relevant labs around the globe.