An interesting embarassment of riches, Dave. Congratulations on your engagement!
K-1:
-US citizen's absense will not make a difference in her ability to petition.
-She will need to get some US income going at or above the poverty guideline--she'll need this for your interview or your eventual adjustment of status.
-You will need to have permission to remain in the UK through completion of your case; you may have to show proof of that to have your case accepted in London.
-No way a K-1 visa is going to get thru in 4 months--it is the most outside chance & better off to plan for closer to 6 months and be happy if something happens sooner.
-You will not have permission to work in the US for the first several months (minimum 90 days after applying for your AOS). Your international travel will also be limited at first.
-Wait time to complete your AOS and actually become a
PR is 2 months to 2 years depending on where you live in the US.
DCF:
-The visa application for DCF is for an Immigrant Visa, not a K-3.
-DCF is done in one visit to NZ--no multiple trips. It also does not take very long--4-8 weeks (perhaps less, contact the Embassy). The US citizen does not have to stay for the whole process, but the visa applicant does.
-The same financial rules apply, but the timeline is moved up. Income/assets must be proved at the visa interview (see form I-864). US citizen must also be domiciled in the US--get a place to live and get some other things on the hoof there (see form instructions).
-Immigrant will have work and travel permission from day one.
-
PR status begins immediately.
-While a longer timeline than NZ, why not DCF from London? With 18 months' residence, her petition may be accepted there. Option to marry elsewhere remains. London DCF is about 4-6 months, US citizen does not have to stay to the visa interview.
DCF is the overal superior choice in my opinion, no matter where you do it.
K-3:
-First find out if you can get married in the UK. Laws have changed regarding marriage, but as 2 non-UK citizens you may be exempt. Factor in length of time to organize this.
-Same issue remains for the K-1 and your UK residency permission.
-K-3 requires marriage + I-130 to the US + receipt notice (takes about a month+ to get) + I-129f petition. Processing times for I-129f vary wildly. Interview timelines for London are pretty quick once you get the peititon approved. No way to complete this in 4 months.
-K-3 can not work without work authorization--takes a minimum of 90 days from application, which can not happen until you are in the US.
Travel in/out of the US may be affected by a pending petition for a K-1 or K-3. There is no guarantee, but if you have good evidence of your trip being business related and a good history of using the VWP, your chances increase.
HTH.
