Possible to break in after PhD?

Literally any comments/suggestions are appreciated. Just want to know my chance and what other people think about my situation.

Have you looked at Equity Research? You skills will be valued more and depending on the group you're in, it's generally more intellectually stimulating. Biotech is a hot space right now and will continue to be for the foreseeable future 

Goldman hire PhDs in London into ER from life science backgrounds every year to focus specifically on life science research so it is definitely possible. 

To make yourself more competitive I would do the following:

• Brush up on financial technicals and valuations so when the time comes you can show you have always had an eye for investing. 
• Adding on from the above point, start familiarising yourself with the operations of the major life science/chemical manufactures business models by reading their annual reports. Having a good understanding of the overall business, combined with the lab-based drug/lifecycle development path can be very useful. 
• Find some ER reports online and start writing your own based off of the companies you have been reading about above. Join your universities investment club and get some mentorship/practice there. 
• Finally, don't pay any of these small firms for internships. Instead try to find time to intern over the summer at a small fund that focuses more on life science/chemical research. If necessary, reach out on LinkedIn to people at the banks you're interested in who are doing life science/chemical research and ask for a coffee with them to better understand their day-to-day roles. 

Good luck!

Do ER, you have industry specific knowledge and skill + allows you to leave to a biotech focused fund at HF

A lot of buy-side biopharma funds take people right from PhDs. Look at Flagship which is more of an incubator at this point but I have seen some at Orbimed, 5AM, and Atlas to name a couple.

Per other comments, life science is hot right now - if you were at a target uni I'd say you'd have a reasonable shot, but I think finance could be tough with lack of experience + lack of brand. 

I would look into consulting as a first step, where your lack of prior experience might not be too much of an issue and there's less resistance to PhDs. I am aware of - just to throw out an example - McKinsey hiring a fifth year medic for a SA position (they converted) from a lower RG uni (not going to name which one) without any real work experience. But while MBB/Tier 2 would be great, I would generally aim for specialist firms - Charles River Associates comes to mind as one of the big players in this space. Networking can be much more effective in consulting, whereas true non-targets are basically stuffed for finance recruiting unless you're diversity. If you can network with someone with a few years at the firm and get a referral, you can often get past the HR screen and then it comes down to your interviews, not your resume. Go on LinkedIn and find someone at a firm you're interested in with a similar background to you and start building a relationship. 

Would also consider that some - generally very competitive - HFs will sometimes take non-quant (but generally still STEM) PhDs if they seem like they can add knowledge for a specific sector, even if they don't have much finance experience. Might be worth shooting your shot for some of them, but do be aware that 90% of the PhDs they will hire will be Oxford/Cambridge/Imperial and maybe another 90% of those will be something very quant heavy. 

EDIT - two other thoughts...

1. Someone mentioned this above but try small bio/pharma-focused VC funds for a first experience. Even if it's a research role, I'm sure they'd love your background and no HR to block you for being at a non-target. 
2. Try to get some sort of visiting research placement in another PhD group at a target uni - a lot of universities (not just your traditional UK targets but also US too) are open to the idea of having students from other unis visit their group for six to twelve months.