`DendogramPlot` accepts `Axes` as an option. 

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

![enter image description here][1]

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}]}}]

![enter image description here][2]

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] &    

![enter image description here][3]


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}]

![enter image description here][4]


  [1]: https://i.sstatic.net/bCCRR.png
  [2]: https://i.sstatic.net/qRMQ1.png
  [3]: https://i.sstatic.net/eTTCG.png
  [4]: https://i.sstatic.net/WA3dX.png