The Rector, Building Manager, and the Conway security guards can provide basic first aid.
For minor ailments, if you are not sure if you need to see a doctor, or for quick expert advice, go to a pharmacy. Pharmacists in the UK are very well trained and can give expert advice on medicines.
The nearest pharmacy to Conway Hall is in Waterloo Station and is called Boots. There are plenty of Boots branches in London, please visit the website to find opening hours and the most convenient branch for you.