Migration with two Apple ID's

You can't. There is no merging of Apple ids.

IF you can't get access to that old apple id, then you will need to download those apps with the new apple id.

IF you did this Change your Apple ID - Apple Support instead of creating a new one, your apps would've transfer over.

Have you gone to the email provider's web site to see if you can find a reset password option?

If you need your Apple ID password, enter the Apple (iCloud) ID and click Forgot Password.

Apple ID - Manage            

Apple ID – Forgot Password            

You will need to recover the password associated with you old Apple ID.

If you forgot your Apple ID password https://support.apple.com/en-us/HT201487

You also can use Having trouble signing in/iForgot https://iforgot.apple.com/password/verify/appleid 

And you will need to see If you forgot the answers to your Apple ID security questions https://support.apple.com/en-us/HT201485

Before trying to answer your security questions is very important that you think about your questions, including your birth date, and their answers carefully and try to recall all details you can. If you don't answer them correctly enough, then you easily can lose control of your account.

You can't change the old ID to the new ID (make them into one ID), but you can remove the old ID, and use the new ID on the old iPad.

When changing to the new ID, there may be some data loss, but you can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos and/or in Photos/Preferences/iCloud. Then make sure full resolution photos are on the device/computer. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion. Also set up Family Sharing  between the IDs to keep access to your apps/purchases. 

 How to switch Apple IDs on your iPhone or iPad.      

How to transfer all files from one iCloud account to another.       

What to do after you change your Apple ID or password - Apple Support.      

Family Sharing.         

Family Sharing - Set up.        

Thank you I did manage to reset the Apple ID password on the old iPad.

Now I still want to change the Apple ID on the old iPad to the new Apple ID I have set up for the new iPad. When I try to do this on appleid.apple.com, Edit/Change AppleID, I am prompted to put in a new email address to use as my Apple ID, but then tells me that new email address is not available? Why is this? Is there a work around?

Thanks again.

Yes, I already created the new Apple ID using my current email address, to use on the new iPad and enable me to transfer data and apps via the iCloud backup. I don't want to create yet another Apple ID. Can I delete the one I created a few days ago?

"Now I still want to change the Apple ID on the old iPad to the new Apple ID I have set up for the new iPad."

You won't be able to do that. The Apple ID you want to change to already is in use.

I understand, but you won't be able to to change the Apple ID on the old iPad to the new Apple ID.