antoniamole

The visa name will have to match the passport name. So you can either change your passport, or apply in the maiden name. You can get a GC in your married name later down the line (when you file to remove conditions, something you do when the initial Greencard expires).
You can still visit while the visa is processing, in accordance with your tourist privileges.

This is wrong.
OP: Oh dear, already a lot of wrong information in this thread.
You can come and get married in May. You do not need a special visa to get married. However, you cannot move to the US on that trip. You can come and get married and then leave, as you had planned to do, in accordance with your tourist privileges.
Then you can start the spousal visa process. First, the US Citizen will file a petition that establishes a marital relationship between you two, then the petition will be approved and do some stuff at the National Visa Center (NVC), then you will apply for a spousal visa in London. Once approved, you will enter the US and you will become a Permanent Resident upon entry. That whole process will take 8-12 months.
Here is the VJ guide for spousal visa. http://www.visajourney.com/content/i130guide1
Good luck.

No problem It is very confusing. Anyone can get married on the US. It's totally fine. It's just that people can't have a plan to marry and then stay on a tourist visa.

I believe from what I've read that the K1 processing time is faster, If you marry on the visa waiver you have to return home regardless and you won't be able to travel to the US while they process your application. (which ive read can take over a year) I'm sure people have stayed while their applications were processed but I can't imagine they are happy about it.
I travelled twice while my K1 was being processed without any problems to the US, both trips were 90 days, I seem to like push my luck,

I would not advise it in her case unless it offered some considerable advantage over the k1 visa. Based on her country of origin and her fiance’s occupation I do not believe it will. However, I can assure you that a legal marriage presented before a court is a legally binding contract irrespective of national origin and/or legal status. A US marriage license trumps many immigration laws. Again, it is all relative to the case….in this instance NO…. in a case of an illegal immigrant from Latin America…..ABSOLUTELY YES!!!!

~Moved from K-1 Process to What Visa Do I Need Forum~
~Inquiry about fiance(e) vs. spousal visas~

Heres a couple of guides,
http://www.visajourney.com/content/k1guide
http://www.visajourney.com/content/k1guide
Should make it easier to get your head around the processes, start with the first link which explains eligibility and how to file and what to send, the second link explains the process after you have been approved.
You could actually start the process before he proposes, we did. Or you can at least begin getting the relevant forms filled out and collecting your photos and other evidence of meeting etc. I think a lot of people decide to get married as the best option of being together and begin filing. I proposed nine months after we began the process, just as well she said yes!

So.....you're advocating committing visa fraud?
OP: Do NOT follow this advice.

Yes, right now K1s are being processed much quicker than IR1/CR1s (spousal visas).
Both K1s and IR1/CR1 applicants can attempt to enter the US while their applications are being processed. It will be up to CBP to decide whether or not to let them through.
It is considered fraud to enter the US on the VWP with the intention to marry and adjust status. AOS from VWP/tourist visas is legal for people who did not come with intent to adjust status (e.g. someone who entered for a short visit, then learned they lost their job back home and decided to marry and stay instead). They will not be able to leave the US until they obtain either advance parole or a green card, or their application will be deemed abandoned.

1. Yes, it is the K1 visa you need.
2. Follow the guides the previous poster linked...not all the forms you listed need to be sent in with the initial I-129F petition. You don't need the DS-156 or DS-230 at all for a K1 -- at least not anymore.
3. There is no set amount of time you need to have known your bf/fiance, the requirement is that you have met in person at least once within the last 2 years. If you have seen him in person in the two years prior to this, you could actually get started on the I-129F petition tomorrow. A formal proposal is not required.