Car starts making weird noise, but only on journeys longer than 20 minutes...

I have a Honda CRV about 12 years old. I normally drive to and from work which is a less than 10 minute journey. On the odd occasion when I need to drive further after about 20 minutes a high pitched screeching/singing noise comes from the rear axle. It only makes this noise when in motion (i.e. not when stopped) and the noise is relative to the speed of the vehicle (i.e. at slow speeds the noise is longer and less often, at high speeds it is shorter and more often). At very slow speeds there is also a grinding noise occasionally.

I've looked underneath and can't see anything obviously broken or missing - but to be fair I'm a complete novice so could be missing something.

Any ideas? I'm due for an MOT in about a month, but wondering if I should be more worried!