Hi Dee! It depends where you're coming into... Is it the UK? If so then it's usually 4-8 hours after they land that you can collect due to the animal arrival reception Team having to process their paperwork etc.
Our boy landed 7am and we collected him at 1pm on hassle. I did have to ring up and check at about 8am though once they'd received him to check the processing time as it varies day to day.
That part is out of control of the carrier company as it is has to do with customs etc.
I would recommend taking at least digital copies of your paperwork with you in case there are any issues with theira getting lost on the crate.

Our boy had a very long journey but he was alright in the end. It appeared as though theyd taken him out of the crate for a bit when he'd arrived as they washed it out and bagged up the mat he was on. He was very dehydrated as he doesn't do well with stress and probably didn't drink, but he slept solid for about 3 days and was right as rain.

Best of luck to you both! Rene