You may be thinking that it's too late to plan your summer international internship, but that is where I'll stop you. Do I recommend you start planning your international internship in May? No, I would not recommend it but at the same time it does not make it impossible. It just means that you have a busy month coming up.
My first recommendation is to plan for your internship to start mid to late June. This means that you can apply for the internship and visa and still have 6+ weeks to get organized.
If you find that your dream internship is in London, you just got a bit more work ahead of you. First you will want to apply for your certificate of sponsorship through BUNAC ASAP. You do not need to have an internship planned to start your application, but you will need to have it secured for them to give you your sponsorship. There are other sponsorship programs but BUNAC is the quickest and easiest, though it does come with a $755 price tag. You will be able to talk with a BUNAC representative so be sure to tell them you are on a deadline. They will work with you to get your application processed quickly.
After you get your certificate of sponsorship you will have to apply for a Tier 5 Visa. BUNAC will provide assistance with the visa application but make sure you expedite your visa application since it can take upwards of 8 weeks otherwise.
Talk with your internship in London and see if you can begin working for them remotely while your visa processes. That way you are slightly familiar with the team and projects before you enter London on a later date.
If you find an opportunity in Dublin you are in luck! Unpaid internships in Dublin do not require visas if they are for less than 3 months. This means that all you have to do is buy your plane ticket and book your accommodation!
For a comprehensive list of things you should plan before going to the other side of the pond check out our checklist made just for you!