How can I reduce the number of nested if statements in this code?

I assume that the code in the question is a method. If not extract it out to a method.

1. boolean appUpdatesAccepted = Parameters.load(mContext).getUpdateType() == UpdateType.BOTH
           || Parameters.load(mContext).getUpdateType() == UpdateType.APPLICATION;
   boolean paramUpdatesAccepted = Parameters.load(mContext).getUpdateType() == UpdateType.BOTH
           || Parameters.load(mContext).getUpdateType() == UpdateType.PARAMETERS;
   

   I'd start with removing some duplication here.

   UpdateParameters updateParameters = Parameters.load(mContext);
   
   ...
   
   final UpdateType updateType = updateParameters.getUpdateType();
   boolean appUpdatesAccepted = updateType == UpdateType.BOTH
           || updateType == UpdateType.APPLICATION;
   boolean paramUpdatesAccepted = updateType == UpdateType.BOTH
           || updateType == UpdateType.PARAMETERS;
   

   I'm not familiar with Android, so UpdateParameters and UpdateType types could be something else.

2. If you reorder the boolean variables you can find some more duplication:

   final boolean appUpdatesAvailable = availableUpdates
       .containsKey("application");
   final boolean appUpdatesAccepted = updateType == UpdateType.BOTH
       || updateType == UpdateType.APPLICATION;
   
   final boolean paramUpdatesAvailable = availableUpdates
       .containsKey("parameters");
   final boolean paramUpdatesAccepted = updateType == UpdateType.BOTH
       || updateType == UpdateType.PARAMETERS;
   

   This could be extracted out to a method:

   private boolean isUpdateAvailable(
           final Map<String, String> availableUpdates,
           final UpdateParameters updateParameters, 
           final UpdateType updateType, final String updateKey) {
       final UpdateType availableUpdateType = updateParameters
           .getUpdateType();
       final boolean appUpdatesAvailable = availableUpdates
           .containsKey(updateKey);
       final boolean appUpdatesAccepted = 
           availableUpdateType == UpdateType.BOTH 
           || availableUpdateType == updateType;
       return appUpdatesAvailable && appUpdatesAccepted;
   }
   

   Usage:

   final boolean tryApplicationUpdate = 
       isUpdateAvailable(availableUpdates, updateParameters, 
           UpdateType.APPLICATION, "application");
   final boolean tryParametersUpdate = 
       isUpdateAvailable(availableUpdates, updateParameters, 
           UpdateType.PARAMETERS, "parameters");
   if (tryApplicationUpdate) {
       ...
   } else if (tryParametersUpdate) {
       ...
   } else {
       ...
   }
   

3. It's also possible to extract out the first two if blocks to a method, they are very similar to each other:

   private void downloadUpdateIfConfirmed(
           final boolean shouldConfirm, 
           final Map<String, String> availableUpdates,
           final UpdateType currentUpdateType, final String updateKey) {
       final boolean shouldInstall = shouldConfirm ? 
           confirmInstallUpdate(currentUpdateType) : true;
       if (shouldInstall) {
           downloadUpdate(currentUpdateType, availableUpdates.get(updateKey));
       } else {
           gotoNextActivity();
       }
   }
   

   Usage:

   if (tryApplicationUpdate) {
       downloadUpdateIfConfirmed(shouldConfirm, availableUpdates, 
           UpdateType.APPLICATION, "application");
   } else if (tryParametersUpdate) {
       downloadUpdateIfConfirmed(shouldConfirm, availableUpdates, 
           UpdateType.PARAMETERS, "parameters");
   } else {
       ...
   }
   

4. I'd move the shouldConfirm flag into the method (to reduce its scope, see Effective Java, Second Edition, Item 45: Minimize the scope of local variables); use guard clauses; create another method which calls gotoNextActivity() to remove its duplication:

   public void update() {
       doUpdate();
       gotoNextActivity();
   }
   
   private void doUpdate() {
       mContext = this.getApplicationContext();
       final UpdateParameters updateParameters = Parameters.load(mContext);
       final UpdateMode updateInstallationMode = 
           updateParameters.getUpdateMode();
       if (updateInstallationMode == UpdateMode.MANUAL) {
           return;
       }
       lblStatus.setText(getString(RString.checking_for_updates));
       final Error outUpdateError = new Error();
       final Map<String, String> availableUpdates = 
           checkForUpdates(outUpdateError);
   
       final boolean tryApplicationUpdate = 
           isUpdateAvailable(availableUpdates, updateParameters,
               UpdateType.APPLICATION, "application");
       final boolean tryParametersUpdate = 
           isUpdateAvailable(availableUpdates, updateParameters,
               UpdateType.PARAMETERS, "parameters");
       if (tryApplicationUpdate) {
           downloadUpdateIfConfirmed(updateInstallationMode, 
               availableUpdates, UpdateType.APPLICATION, "application");
           return;
       }
       if (tryParametersUpdate) {
           downloadUpdateIfConfirmed(updateInstallationMode, 
               availableUpdates, UpdateType.PARAMETERS, "parameters");
           return;
       }
       lblStatus.setText(getString(RString.app_up_to_date));
   }
   
   private void downloadUpdateIfConfirmed(
           final UpdateMode updateInstallationMode,
           final Map<String, String> availableUpdates, 
              final UpdateType updateType, final String updateKey) {
       final boolean shouldConfirm = 
           (updateInstallationMode == UpdateMode.CONFIRM);
       final boolean shouldInstall = shouldConfirm ? 
           confirmInstallUpdate(updateType) : true;
       if (shouldInstall) {
           downloadUpdate(updateType, availableUpdates.get(updateKey));
       }
   }