What is the quickest way for my husband to enter the US?

Ideally, the quickest way to be reunited with your husband is for him to enter the US on a K-3 nonimmigrant visa so that he he can bypass the lengthy processing of immigrant visas. You can accomplish that by submitting the traditional I-130 petition for your husband and then submitting a separate petition for him on form I-129F immediately after receiving a receipt for the I-130 petition. Once he enters the US with a K-3 visa, he can complete the immigrant process by applying for adjustment of status.

Well, there isn't really the quickest way. The best way is to file form I-130 for him, but that processing will take time. Another proper way would be a K-3 visa, but then you would have to do an adjustment of status for your husband, so the overall total cost of the process will be more and it will take longer. So do an I-130 for him. If he enters the U.S. on a visa waiver, (1) he could be denied entry if the immigration officers learn that he has a US citizen wife in the States, and (2) the intent is an issue because visa waiver program is in place only for temporary visitors who have every intention to leave the U.S. and go home.

It's a bit complicated. If he came to the US to visit but then changed his mind, that could work. But you would need something happening in the US that made him decide to stay during that particular trip. And he would need a round trip ticket when coming into the US. It has to look like a visit. Once here, you should wait a month or two before filing. The entire process only takes about 3 months from start to finish. You should consult with an experience immigration lawyer before making any plans.

If he is in the U.S., he may apply to adjust status. If he is abroad, you can start immigrant petition. You may also consider applying for K-3 if the wait is long.

You petition for him and it takes about 9-12 months for him to get a visa at the US Embassy in London after that. Otherwise, he can try to enter as a visitor and you wait for about 3-4 months and then file for him within the US. I can discuss the process in more detail with you if you like to explain the issues and strategy.

I assume that you want to petition for your husband so that he can get a green card. The two options are consular processing and adjustment of status. Since he is outside the U.S., he would consular process. Since he is an immediate relative, the entire process may take 6 to 9 months; possible even less. However there is no way to guarantee how long the process will take. Consult with an immigration attorney to guide you through the process.