What type of visa should I apply for?

I assume you're on Tier 4 currently? You can work full time outside of term time so if it's in the summer it won't be a problem. During term time it's 20 hours a week. You won't need another visa and you can only have a single visa at a time. Applying for a different visa would affect your status at medical school ad your tier 4 would become invalid, though technically there are no eligible visa options you could apply for anyway.

I hope this helps. Also stop into your university international department and get something in writing to confirm the hours you can work to help cover your back.

I'm studying medicine outside the UK. I arranged the clinical placement with the hospital directly (not via a university). Will I still be eligible for Tier 4, or I should check other options like "Student visitor" or "Doctors and dentists"?

The most appropriate visa is the Tier 4 General. This is typically for students but the government requests those taking medical electives to apply for this. This would require you obtaining a CaS letter (number) from the institution. If they are not registered to sponsor tier 4 then forget it. Usually it is only educational institutions that sponsor tier 4.

A certificate of sponsorship is for Tier 2 visas. A registered tier 2 sponsor issues a number to you (this number is your certificate). There is no physical certificate only a virtual number. The employer applies online and obtains one providing they have some quota left. This will not apply to you as you don't qualify for Tier 2.

If you are not an EEA citizen then you require a visa for the placement. If the hospital let you attend without a visa then they would be breaking immigration rules as would yourself which could have repercussions. Double check with the hospital hr staff as they may assume you are in the UK studying medicine already with a tier 4 visa.

Don't go to your local British embassy as you will be rudely advised by security to check their website.

I have read the Home Office website information and understand the rules.

The Tier 5 visa categories are typically work related (unless charity/religious). As clinical placements would be unpaid they would not qualify under Tier 5 as the national minimum wage needs to be paid. The Tier 5 government exchange scheme usually require require applicants to be graduates and the only medical placements offered are from MTI which is for medical graduates. More can be seen in appendix N.

The Royal Brompton & Harefield NHS Trust is not a registered Tier 5 sponsor either.

I would advise the op to contact the hospital hr dept. They are likely associated with a London medical school (Imperial I think) which would provide an option to go through the Tier 4 route.

Just had a quick look and it looks like you might need this one. But you'll need to see if the hospital can issue you with a certificate of sponsorship for that (see the link). The invitation letter they sent you may well be the equivalent of this. I suspect you'll have to go down to your local British embassy and see what they advise.