There is more or less no "quickest way".
You anyhow will need to (1) apply for national visa (which will be processed by both embassy and immigration office, i.e. Ausländerbehörde), (2) enter Germany on that visa and start working, and (3) then apply for residence permit (e.g., Blue Card).
Answering other questions of yours:
up to 3 months to get ...
I had the same question and emailed Bundesagentur für Arbeit (BA). In my case, the officer said that, with the internship
less than 90 days within 12 months, and
during my study after 4th semester in H+ institution
there is no need for BA's 'Einvernehmen' because it is not considered an employment (BeschV § 30 Nr. 2;...
H1B may not be the right visa type for you. Instead, I suggest reading about, and discussing with your employer, the L1 visas One of them may fit your situation. They are designed to allow companies with international operations to move overseas employees to their US operation.
I see two major hurdles which you'd need to overcome.
Firstly, you would need to convince the company that the not-insignificant expense in both their time and money would be outweighed by having you working on-site in their office in the USA instead of remotely from London.
Secondly, the company would have to convince the US Dept of Labor that there are no ...