I had to get PVG/Disclosure checked for my job, and was worried about it being delayed. Turns out, according to my boss, it took less time for the to clear my US record than it took to clear hers from England & Wales! So I really wouldn't stress about that.
As for a Criminal Records check, are they asking for something specific? I.e., do they want something from the FBI, or from each local police dept where you've lived for the past x years? Find out what they want, specifically, and just let them know that the more different agencies you need to contact, the longer it might take.
As for making actual requests for records... I'd just google it. The process is likely to be different in each department, but probably not actually difficult. Just time-consuming.
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