UKLDNChris

Hi,
Yes, my first AP expired Dec 24 2018. I filed for renewal and got an NOA1 date of October 1. My concern is more around whether my trip back to London in Nov would affect the pending AP application.
Thanks for your updates - glad you got the GC. Unfortunately, I think things are moving exceptionally slowly in NYC. 

It'll probably depend on the CBP agent who deals with you on each trip. On the first few there probably won't be an issue and they will have no idea that you have filed a K-1 although if they ask Do Not Lie!! As time goes on you might get a curious one asking what you do for work and how you are able to travel so often and again you must be upfront as any perceived deception will be seen as fraud. When I made my second 5 week visit to Florida in the space of 15 weeks the CBP was very curious about how I got so much time off work. 
 
Also the ESTA program is probably not really meant for working while travel even if that is 'working from home' and you are employed by a UK company so it'll depend on how it's viewed by the CBP agent you see.
 
Good luck.
 

Never lie to immigration, but I'm unsure of the legality of all of this.

frequency of the trips should not be a problem. you already have a record of coming and going on time.
 
visiting clients is not a problem. since they are already your clients and you have worked with them before.
 
working from home while in new York may be an issue 
 
good luck