I recently finished my Ph.D. in the US, and after so much struggle I've found a job in a startup in the UK as a scientist. The salary offered is the annual salary of GBP 40,000 + GBP 3,500 for relocation to the Cambridge area. However, the visa fees + IHS fees have exceeded well above the relocation allowance provided. In fact, I needed to dip into the minimal savings I had while doing my Ph.D. to fund my relocation. I would like to ask for an increase in the relocation package. How would be the best form to approach this?
First there is the reality check, and then there are the employable methods.
While a "signing bonus" in the form of a "relocation package" is not unusual, it really is just meant to facilitate your decision towards a "yes". Which you apparently have already given. Your negotiation upper-hand is gone and now you start your actual relationship with that company. In general, asking for an advancement upon starting a job is frowned upon, but you may check if this is culturally applicable to Cambridge.
When assessing a job offer, you should think like a contractor who is quoting his/her services. You should have calculated what your moving expenses would have been and negotiated the relocation package on this basis instead of accepting whatever they've offered you. Your "quote" cannot be correct if it does not cover your costs of operation (or moving in this case) and costs of opportunity (usually previous salary and benefits).
Also, I'd normally say that any kind of signing bonus is just a deal sweetener, not the actual meat of the deal. So it would be totally normal if the company informed you during negotiations that the "relocation package" is to alleviate your financial burden of moving but not to ultimately compensate each and every penny you would spend on it.
Having established the earlier points, you can ask for an additional "relocation package", but I would strongly advise that you give a concrete number for the company, specifying how much you need and for what. Preferably, attach proof of your current expenditures and the market costs for the upcoming ones. This way you prove that you've done your homework this time and there will not be a second request for extra package.
Consider as well getting a loan from a financial institution. In Europe, some places have very reasonable rates for credit lines that seem pretty applicable to you, maybe in the US as well. If you get denied due to score credit or something similar, then consider attaching this to the aforementioned request for an additional relocation package, as this is further proof that you may not have the means to actually relocate unless they were to help you.
As a last resort, consider asking friends and family for help. Explain to them that you will be living frugally for staters (and do plan to do this if you are getting a loan no matter if it comes from your parents or from a bank) and that you have a financial planning for the upcoming months. Give them clear numbers, so that you can tell people how much you need to gather in total and how long you will take until you repay everyone. Consider paying a 5% yearly interest rate (as a courtesy, because at this rate and risk, people are doing you a favor, not investing), and be ready to sign a contract on paper about whatever debt you may incur.
Also, consider selling some stuff of yours, even if at a low price in case the amount you need is compatible with your current possessions. Textbooks, action figures, old clothes, a heater, and many other items that may be hard to take across the ocean are good candidates for cash conversion.