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DendogramPlot accepts Axes as an option.

Inter-cluster distances in a Cluster object are given as the third from last element.

In the following, these distances are highlighted in red:

Needs["HierarchicalClustering`"]
Grid[{{Agglomerate[{1, 2, 10, 4, 8},
 DistanceFunction -> Automatic, 
 Linkage -> "Single"]}, 
 {DendrogramPlot[{1, 2, 10, 4, 8},
 DistanceFunction -> Automatic, Linkage -> "Single", 
 LeafLabels -> (# &), ImageSize -> 300, Axes -> {False, True}, 
 AxesOrigin -> {0, Automatic}]}}]

So ... this verifies that vertical axis does indeed measure the inter-cluster distances for a given DistanceFunction and Linkage.

For various combinations of DistanceFunction and Linkage you get the following pictures:

{#, Agglomerate[{1, 2, 10, 4, 8}, DistanceFunction -> Automatic, Linkage -> #], 
 DendrogramPlot[{1, 2, 10, 4, 8},
 DistanceFunction -> Automatic, Linkage -> #, 
 Axes -> {False, True}, AxesOrigin -> {-1, Automatic}],
 Agglomerate[{1, 2, 10, 4, 8}, DistanceFunction -> ManhattanDistance, Linkage -> #],
 DendrogramPlot[{1, 2, 10, 4, 8},
 DistanceFunction -> ManhattanDistance, Linkage -> #, 
 Axes -> {False, True}, AxesOrigin -> {-1, Automatic}]} & /@
 {"Single", "Average","Complete", "WeightedAverage", "Centroid", "Median","Ward"} // 
 Grid[Prepend[#, {"", "EuclideanDistance-Clusters", 
 "EuclideanDistance-Dendogram", "ManhattanDistance-Clusters",
 "ManhattanDistance-Dendogram"}], 
  Dividers -> All, Alignment -> Bottom] &    

EDIT: Note: Despite syntax highlighting in red of Axes and AxesOrigin, the options seem to work:

DendrogramPlot[Prime[#] & /@ Range[30], Axes -> {False, True}, 
AxesOrigin -> {-1, Automatic}]