Joining Twin Waffle Weave Blankets?

There are several different ways to do this, but some of the differences will be whether you will be using a sewing machine or sewing this by hand.
Your picture is not very clear as to what a 'waffle weave' blanket may look like, so here is a link to show what you're asking about.
www.amazon.com/.../s?k=Waffle+Weave+Blankets (Affiliate Link)

First; check out some of the suggestions from similar questions. Be sure to check each one as you may see something you could use.
www.thriftyfun.com/tf80294321.tip.html#feedback
www.thriftyfun.com/Use-Two-Twin-Blankets-to-Make-a-King...

I cannot tell if your Mother's blankets are new or used, so I would like to add some hints on how to care for waffle weave and what needs to be done before starting a project like this.

"Use cold water, not warm or hot. Cold water will help prevent the blanket from shedding. Use the gentle cycle on the washer. This is especially important in top load washers as the agitation bar can act as an abrasive on the blanket, causing it to shed.
A quick tumble on low heat afterwards will soften them even further. Also, always use medium or low heat if you machine dry - high heat can cause cotton fibers to tighten, making the fabric feel stiffer.
Shrinkage. Because waffle weave is 100% cotton and woven in a flat weave shrinkage on the first wash is expected, shrinkage on first wash is approximately 18% in the length and width."

I have seen similar projects completed, but while each one is similar, none will be the same, so suggestions will have to be adapted to your individual project.

First; you will always have a 'strip' down the middle of your blanket. This can be irritating to some people. There are several types of material that can be used but you should try to select material that is soft but durable. I would suggest that you visit one of your craft stores and explain what you're planning to do. A clerk should be able to show you several types of material that comes in finished form; usually in rolls.
They may also have some suggestions for how to proceed.
Most people opt for satin blanket binding similar to this:
www.etsy.com/.../2-white-satin-blanket-binding-25...

How you attach this to join your blankets will depend on whether you are doing this by hand or with a sewing machine.
This is a tedious and slow project no matter which method you use.

If I was going to tackle a project like this, I would ask my questions from experienced 'sewers' so when I ran into a problem, I would have a place to ask more questions.
The members in both of these groups are very good with all types of sewing projects, so do not be put off by the names.

www.quiltingboard.com/
sewing.patternreview.com/.../fuzzysearch.pl?phrase=&st=t&name...