Sorry to hear this. Of course we understand, our cats are our families too so we can relate very well to what you are going through.
However, you are in a uniquely difficult position and perhaps need to talk to your doctor as the anxiety is likely to spiral when your partner goes away.
There's a lot you can do that will be more productive than flying into a flat panic and running out of the house, and may serve to calm you as you'll know you've taken every possible step to get your cat back home safe. Take a moment to think about what you could do if you had found a lost or injured cat, then work out what you need to do to find that person.
Make sure there's fresh, smelly food outside the house each morning and evening. If it was Basil eating the food he would present himself how he normally does, so I doubt it is him.
Cats have excellent sense of smell, better than dogs. Tie one of your well used, unwashed t-shirts or night dresses out on the gatepost or washing line, your scent may bring him out of hiding. Post loads of your flyers on your gate, all passers by need to be aware there is a missing cat and he lives here.
Gill's advice to take your mobile phone when you go out is very wise. I'd go a step further - if you're likely to dash off regularly, why not pack yourself a search kit?
Being prepared can take some of the panic away and help you stay safe. You'll need a mobile phone with full charge and emergency numbers already saved to memory, a good torch, a water bottle, chocolate bar, blanket or shawl in case you get cold (or if you find Basil you'll need to wrap him securely) a box of cat biscuits to shake, gloves, a French phrase book in case you get stuck, some cash for a taxi to the vet (or to use a public phone box) and your credit card for emergencies.
Take a photo of Basil with the words "Basil, Il est perdu - aidez Moi?" (He is lost - help me) written on the back to show any strangers who want to know why you are on their property.
Make sure you wear a bright coloured coat or hat, so traffic can see you in the dark.
When searching do take time to stop, listen hard, call softly. Don't panic just take a few steps at a time then call him again. Shake the biscuits, but the most important thing is to listen hard. If he's stuck in a building his cries will be hard to hear so be sure to walk all round buildings listening for him.
Post notices in the local shops, pub, library, school, police station and of course the local vet, if he's been picked up by someone you need to make sure they see your notice. Also there will be an internet site for locals similar to Gumtree that you will want to post a "lost pet" notice on, that way if someone is surfing the internet trying to find his owner there will be a hit in your area for a lost cat with his description.
When your partner is away, make arrangements to telephone each other at a specific time each morning and evening, then if anything happens to you he will know there's something wrong, and can take steps to get you some help. You're going to need some support, is there a Women's Institute or other social group you can join? Maybe a yoga class, gardening group or supper club to give you a distraction and maybe make some friends who can help support you.
Do take great care, and be sure to come back here, we can't be there for you but we surely can be here.