London undergrad unstructured internship - cold email (direct ask) or coffee chat/network?

Few points on this, albeit not that coherent:

• Most people won’t do a summer in first year, the spring weeks are enough. However, if you can get one it does confer an advantage.
• Assuming you’re at a target, there are usually finance-related (may be in consulting but you can get modelling experience at a lot of boutiques) internships readily available through careers portals. A lot of these are ad-hoc, unstructured internships and they’ll likely advertise only at a few schools.
• Message alums on LinkedIn, that’s your best bet to find stuff entirely cold.
• Ask people in the year above who’ve done first year summers for ins. It’s pretty common for different departments to have a few small (normally no name and local) firms that always take one or two interns informally from that same department each year; knowledge is passed on from one cohort to another re. recruitment and openings. 
• Consider academic RA positions at your university and others, anything is better than nothing and you’ll get paid. If you can tie it into something relevant to an industry you’re passionate about, all the better.