Build in the extra time your transport method specifically requires: at least 90 minutes before a Eurotunnel Le Shuttle departure for Pet Reception check-in, the standard airline recommendation for a flight with a cabin pet, or simply earlier than you think for a ferry crossing with pets aboard.
Keep your pet's documents — EU pet passport or health certificate, and the category permit if it applies — in the car or your carry bag, not packed away in checked luggage or the boot under everything else. A border or train check that can't see the paperwork immediately causes exactly the delay you built in extra time to avoid.
Before you set off, note the nearest 24-hour vet clinic or regional emergency rota number for your first overnight stop in France, and keep it saved somewhere that doesn't depend on a single device's battery. It's a five-minute step that matters only if something goes wrong — which is exactly when you won't want to be searching for it.