Does anyone know whether I should choose: nearest, bilinear, cubic or majority when projecting the WorldClim altitude raster layer?
Michael Miles-Stimson is correct in his comment above; nearest-neighbour (NN) and majority resampling methods should only be applied to categorical data, i.e. nominal and ordinal level data. Elevation, even when it is presented as integer values (which is a practice that I wish we could make illegal and punishable by lengthy jail terms), is not categorical. Elevation is ratio level, e.g. an elevation of 20 m asl is twice as high as an elevation of 10 m asl. Elevation is also a phenomenon that is a continuous variable, i.e. 10.432467533 m is perfectly legitimate as an elevation (darn those integer valued DEMs!). Therefore, you should be using bilinear (BL) or cubic convolution (CC) resampling methods when dealing with these data. The difference between BL and CC resampling is essentially that BL resampling will be slightly faster and result in slightly less smoothing of the resulting surface because CC interpolates the output value using a greater number of input values within a local neighbourhood. Both methods are however good options for elevation data.
I should note that this is not simply a preference. The inappropriate use of NN or majority resampling on continuous variables like elevation will result in subtle, sometimes difficult to discern, artifacts resulting from the duplication of rows/columns at regular intervals in the output image. Here is a very good example of the kind of artefacts that can result in a DEM when choosing NN inappropriately. Notice that in this case, the artefact was only noticeable in a derivative of the DEM (curvature) and not the DEM itself. Often it becomes apparent that something isn't quite right with your NN-resampled DEM when you create a hillshade image.