I recall that older versions of SourceTree used to handle submodules automatically, but it does not in more recent versions.
There does not seem to be any manual way to update submodule in SourceTree either.
This is particularly painful when there are many nested submodules.
The only way I know of updating submodules in SourceTree is to manually open each submodule (and recursively open each nested submodule) and look at the "Uncommitted changed" select only submodules that appear to have change and "Reset" the changes.
It would be nice to have options to update submodules when cloning, pulling, and switching branches. A checkbox in each of these dialogs would be sufficient.
At minimum there should be a way to manually update all submodules recursively. A warning notifying you that submodules need updating after performing an operations would also be useful.