free software - How could I turn my data into a heatmap / intensity map?

I have database that tracks a large number of entities. Each entity has a 'coverage area' or 'catchment area' - a radius in miles from a specified UK postcode, eg

Leyton office     4.3m    E11 4LL
Stratford office  5.2m    E15 6ZZ
East Ham office   2.1m    E4 8QQ

I want to turn this into a geographic intensity map - a map of a region which is shaded or coloured to indicate the number of entities that cover each point on the map. Segmentation into small regions is also acceptable.

This is probably a one-off project, so it's unlikely my organization will provide money to buy software. That means zero or at least low-cost solutions are preferable.