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There can be multiple solutions for your question. One such is mentioned here in this answer. Quote:

I once extracted single apps and apps data using a normal archiver explorer in my Arch Linux system. In Windows, 7-Zip FM or Winrar should do the trick.

Obviously it is quicker done if you restore the full backup all at once with a custom recovery (but I believe you already merged the archive parts the backup is done in), but I once made a backup after the system launcher went to a systematic crash, so I couldn't do anything, and doing a full restore would led to the very same problematic, so I had to get back the data of the apps I wanted to restore that way.

Just open the .tar files, they are compressed archives. data.tar contains what the data folder contains in your Internal Storage: upon all the folders, data/app contains the apk of the applications you have installed, and data/data their saved data, which is you may want to restore. Inside that folder, every app has its dedicated folder (e.g., data/data/com.google.android.apps.maps for Google Maps). Copying it in your data/data folder in your phone with a root file manager will restore the data for that app.

Again, for a complete restore, just copy the whole packages (supposing you left them untouched) in the directory they were (to find out which it is, do another backup and you will see where it is created), boot into a custom recovery like CWM Recovery and do a full system restore, pointing to where the backup is located in your phone.

The other method I found is to use an app like Nandroid Manager *Root from Play Store. It can help you extract the apps and the data from the recovery backup. There are several apps available here.

Otherwise, you can always do the following:

1. Take Nandroid backup of new ROM;

2. Restore Nandroid backup of old ROM;

3. Take Apps+Data backup through apps like Titanium Backup or Helium;

4. Restore Nandroid backup of New ROM;

5. Restore Apps+Data backup in this new ROM by using apps from step 3.

Note that this doesn't concern with your data in Internal/External storage. Nandroid backup possibly didn't took the backup of this data and if your new ROM wiped these storage(s), then you need to look for Data Recovery.