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For both AKC and UKC, it takes years and years and years, and MANY generations of dogs past the original F1 generation. You have to get your "new breed" to "breed true," that is, to consistently reproduce itself, true to type. The parent breed club must write a clear, concise breed standard describing everything about the dog. You'd need to form a strong club with many members breeding these dogs, and hold specialty shows, including a breed national show.
Once all this is accomplished, IF the kennel club approves your application, your new breed may be admitted on a kind of probationary status. They will be able to participate in "non regular" classes at shows, and judged against their standard, but not against any other breeds or eligible to compete for Groups or BIS. If everything goes well, after a number of years, the new breed may be admitted as a "regular" breed.
If you just have a certain F1 cross that you feel is great at what it does, then what's wrong with just calling it a "Lab/Border Collie cross?" The made-up names are cute, but without true breed type, they don't really mean anything. I say just call it what it is, do what you want to accomplish with each dog, and have a good time with them.
Or, if a person just dying to register something as a "breed," then there are "registries" out there who will register ANYTHING, based on nothing more than your signature and your $$$. You can call it whatever you want. I've seen "registered" dogs such as "schnorkies," "maltipoos," etc.