What is the average cholorphyll concentration for the ocean?
Behrenfeld and Falkowski, 1997 have maps of cholorphyll concentration for the entire world on pages 11-13 of the linked paper. The maps are divided by into yearly concentration and seasonal concentration, with the maps filled out from experimental data by two different modeling methods. 

These maps are in mg C / m$^2$ / year, but they offer the normalization of $$\frac{4.6 \text{mg C / hour}}{1\text{ mg Chlorophyll}}.$$ Chlorophyll production varies by optical depth, which is the distance that light can penetrate into the water. Optical depth depends on various characteristics of the seawater like sediment node, so it doesn't have an exact analogue to water depth. However, Fig 1A supports a conclusion that plankton production is reduced by an order of magnitude at depth of 20 meters, and by 2 orders of magnitude by 60 meters.

If we confine the chlorophyll to entirely the top 60m of the ocean, then we can convert $$\frac{1 \text{ mg C}}{\text{m}^2\cdot\text{year}}\cdot\frac{1 \text{ mg Chlorophyll}}{4.6\text{ mg C / hour}}\cdot \frac{1 \text{year}}{8760 \text{ hours}} \cdot \frac{1}{60 \text{m}}= \frac{0.41 \text{ ng Chlorophyll}}{\text{m}^3}.$$ 

Using this convertion specifically on map A on page 11 of the linked paper, I get for your answers:


Open ocean in general: 75-150 g C/m$^2$/year $\rightarrow$ 30-60 $\mu$g/m$^3$
N. Atlantic Ocean: 150-250 g C/m$^2$/year $\rightarrow$ 60-100 $\mu$g/m$^3$
Coastal waters around the UK: 300-450 g C/m$^2$/year $\rightarrow$ 125-185 $\mu$g/m$^3$
Equatorial waters: 100-200 g C/m$^2$/year $\rightarrow$ 40-80 $\mu$g/m$^3$