Hello & welcome, I am Dr. B, a licensed veterinarian and I would like to help you with your wee one today.
It isn't uncommon for cats with outdoor access to disappear due to exploring too far, getting lost, and misadventures. This can especially be the case with male cats (even neutered) who are keen to explore their local territory. And it is quite possible for them to disappear for weeks before turning up home, in the local shelter, or vets.
Now it looks like you have been very thorough so far, so keep up the good work. I am glad to see you are putting up fliers and keeping an eye out. As well, notifying (and reminding/checking in with them) the local vets, sanctuaries and councils was a good start as well. That said, I do want to provide a few more ideas and information to helping you find him.
First things first, here is a very good resource for how to hunt for a lost cat is the Cat Chat (http://www.catchat.org/lost.html) website. I would advise having a read over and use this as your plan of attack.
As well, besides physical sanctuaries and shelters, make sure to be reaching out to the small home based cat rescues. Many of these don't advertise or have a physical presence you can visit, but if you speak to those local vets and ask them to provide the contact details of all the cat rescues or charities they work with, you can ensure you aren't missing the one that could have him. This goes for the smaller branches of Cat's Protection (HERE) but also the smaller local cat charities. So, just to make sure you are not missing any of the smaller cat safe houses, you can check via Cat Chat @ http://www.catchat.org/adoption/index.html and your local RSPCA branch (SEARCH HERE). . As well, it is worth a quick scan through your local Gumtree and Preloved websites in case someone has posted there
Furthermore, the internet is a valuable resource and there are a number of databases that people can list lost cats, as well as those who have turned up without an owner. Register him as lost here, and include a photo if you can (it makes it much easier for people to realize who the cat they have seen might be). Check the found cats.
Animal Search UK: http://www.animalsearchuk.co.uk/
UK Missing Pet Register: http://www.nationalpetregister.org/mp.php
Cat Chat’s Database of Registers: http://www.catchat.org/lost.html#national
Finally, if Chico has a microchip to identify him, do let the microchip company know he is missing. They can make sure your contact details are up to date and flag his account should anyone phone about his chip number. This can be invaluable in situations like this to get our kitties home.
Overall, you have done great so far, but the above are a few more steps to take. As well if you haven't already on your last neighbor visit, do make sure to be checking (or asking that they check) their garages, sheds, and cellars for him. Cats do sneak into odd places and can get locked in and trapped. As well, if there are any neighbors have recently moved out and left a house empty, you may want to contact the owner or realtor to have them check that he hasn't gotten trapped in there. But overall, as I am sure you have been told by the vets and other people you have been in contact with, do keep up hope and keep looking. Kitties do wander, but sometimes we can give a little help to getting them back home. It will take a lot of work, but I am sure he is worth it.
Best of luck in your hunt for your wee man.
All the best & my thoughts are with you for Chico's safe return,