We had our 3 month old in the carrier during his nap. We’re concerned that there was a moment where his breathing was restricted, and it took him a while to come out of his sleep. Is this normal and what should we do to check? Thanks in advance

I’m sorry you have had this scary experience. Is your baby ok now?
Please always check with your GP &/or Maternal/Child/Family Health Nurse whenever you are concerned with health/safety of your baby.

Red Nose recommends that all babies are placed on their back for sleep as this is the safest position for baby to protect their airway.
This poster helps to explain this - https://rednose.org.au/downloads/RN0492_Back-is-Safest_Poster_A3.pdf

Red Nose also has information about safely using baby carriers.
Wearable devices are a great way to hold baby hands free, but there are some important safety information which include:
“Dangerous positions include lying in a sling with a curved back with their chin to chest, or lying with their face pressed against the fabric of the sling or the wearer’s body.
Babies who are under four months of age, were born premature or low birth weight, or have breathing difficulties, are at greater risk of suffocation.”......... https://rednose.org.au/article/slings-and-baby-carriers
Please have a look at the entire article, which has recommendations for safe use of baby carriers.
Product Safety Australia also provides some helpful information, including sleeping baby in these devices…..  https://www.productsafety.gov.au/consumers/keep-baby-safe/move-baby-safely/baby-carriers-wraps-and-slings-guide