I am from Chile, 40 years old with 5 years experience in the academic technology transfer. What would be the best way to obtain an invitation from a UK startup to get a short term and non-paid position in London in order to gain experience.
Find start-ups in your field and contact them just as you would if you were applying for any other job.
Make sure you send a CV with a cover letter (letter of inquiry). In the last paragraph, in addition to providing your contact information, put down a date you will contact them to touch base, then make sure you do.
The main concern a company will have is your legal status. Do you plan to go to London as a tourist and "work" for 3 months? Do you need a visa to go to the UK? Do you have proof that you have enough money to live off of? Do you have a residence/a place to stay?
You see, the biggest problem is the UK, just like the rest of the EU, will not be able to issue you a visa or a work permit, regardless of whether you are getting paid or not. They would have to prove they cannot find a citizen of an EU state to do the job, and that may not be the case. Even if it was, and you are the only person who can do the job (and is interested), the company will have to fill out tons of paperwork and pay quite a bit of money-something most start-ups are in short supply of-to have you there.
So, before you send out inquiry letters, you should make sure you have the legal aspects covered.
Good luck, and if you need additional information, please give me a call. I have worked in Europe for over 7 years, and am very familiar with the issues non-EU citizens have when it comes to working there.