How fast can I possibly ship out to air force basic training?

Less than three months is about as fast as it ever gets and that's assuming 100% of everything goes absolutely perfectly. 3-6 months is about average leaning closer to six months. It can take as long as a year if there are any hang-ups at all -- and there frequently are.

If you have zero concern about what job you'll be doing, etc., and are willing to take ANY slot that comes up, the process can go fairly quickly. But ultimately you are limited by the available class start dates. When you go to basic depends on when your tech school class starts. Some tech schools only have one or two classes per year. Some have them starting every couple of weeks. Your BMT date has to align with your tech school start date. That's all contingent on completing the rest of the enlistment process (ASVAB, physical, etc.). So it can take a while to get everything in order.

It's a process and it takes time to get all of the ducks in a row. It moves at its own pace and no faster. There is no one who has any influence over the rate at which that proceeds; not your recruiter or anyone else. You should plan on a six-month wait before shipping out. If it comes earlier, great. But it is not likely to go much faster than that.
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average is 6 months. you might be able to cut that in half by be willing to do ANYTHING an being willing to ship out with zero notice( as in get the phlegm call on Tuesday to leave Wednesday)

probably will be late dec before you could ship out

don't be surprised if it is jan or feb before you ship

Maybe 6 months at the soonest. BMT and Tech Schools slots are being filled at least 6 months in advance. Your only hope is that someone else backs out and you take their BMT and tech slot. You will have zero choice in what type of job you would get. It could be a cook.