How should I systematically go about looking for bursaries/scholarships for an internship in the UK?

In the UK, there is not much funding for non-UK/EU students. There is not much funding for undergrads. And there is not much funding for internships. On top of this, as a non-EU/UK student, you will need a visa to work/study in the UK and unpaid "internship" is not a category for which you can easily get a visa. Tier 5 visa is perhaps most appropriate, but it requires serious commitment from the sponsor (UK university), which they may hesitate to provide.

To summarise, my advice is to look for a more standard route, e.g. an undergraduate course for which you can probably get a bursary from your own country. You can also apply for a paid work employment (e.g. postdoc) if you qualify.