London Analysts Exit Options

Read a lot around the forum but didn't find anything useful - hence decided to start this thread to more systematically assess my options + hopefully useful for some more people in the future.

Currently in a top sector team at a decent MM bank (MM, think RBC/Jefferies/WF) in london. However, I quickly realised that London pay is shit and if i ever had any chance of making money that would be in the US. Similarly, my girlfriend is also currently studying in the US (a further reason for transfer). For context, i am currently 21 so I feel like I can take on a lot of risk + a lot of grit and determination for work.

Ideally, I'd like to exit from banking and get into PE, however, I appreciate how hard this can be. Couple this with the "internal transfer" situations to actually reach the US, I wanted to ask for opinions. 

Key questions:

• Given I just joined (and hence my lack of knowledge regarding how organisations work), when can I expect to submit a transfer request for NY?
• For maximising long term prospects in PE, should I first jump to PE in London and ask for transfer there? I know that US does on cycle recruiting so by the time I get to NY through my bank I'd be too late already
• Any other comments welcome!