Opportunities to find independent funding for doctoral studies are usually more limited than at the postdoctoral level but many countries have their own schemes for funding PhD programmes. You can find out more about opportunities at UCL here, including possible external sources of funding and several schemes are listed on this page below.
There are numerous schemes available that you can apply for to fund a postdoctoral research position. There is an excellent database of opportunities here. UCL also has a list of postdoctoral and fellowship funding opportunities here. Some examples are listed on this page below.