What should I call someone addicted to mobile appliances?

There's a CNN article about “smartphone zombies” that implies zombie is appropriate (perhaps in its sense “A person, usually undead, animated by unnatural forces ..., with no soul or will of his/her own”). The article also mentions Stephen King's novel, Cell:

... the protagonist, Clayton Riddell, doesn't turn into a zombie because he doesn't own a cell phone. The story is about an event called "The Pulse" that turns cell phone users into vicious, mindless beings.

Apparently Nomophobia is the term commonly used for cell-phone addiction. It is an abbreviation of no mo-bile phone phobia

These are people who seldom switch off their cell-phones and start feeling stressed when forced to stay away from them.

It doesn't seem like there is any word for cell phone addiction. But this comes as close to it as possible.

Wikipedia has an article on computer addiction, a fairly broad term that seems to include the situation you are talking about. It begins like this:

Computer addiction is a mental illness which causes the excessive use of computers to the extent that it interferes with daily life. Excessive use may explain problems in social interaction, mood, personality, work ethic, relationships, thought processes, or sleep deprivation. The Diagnostic and Statistical Manual of Mental Disorders does not include a diagnosis for such a disease.

The term computer addiction originated long before the Internet.

Some people develop bad habits in their computer use that cause them significant problems in their lives. The types of behavior and negative consequences are similar to those of known addictive disorders.

This is connected with the related Internet addiction disorder, or IAD:

Internet addiction disorder (IAD), or, more broadly, Internet overuse, problematic computer use, or pathological computer use, is excessive computer use that interferes with daily life. These terms avoid the term addiction and are not limited to any single cause.

That one may be more appropriate, since it zeroes in on the networking aspect more than on the gadget aspect, which I believe is what you were talking about. The term is not without controversy, and does not appear in the DSM-IV. The same article later says:

Online activities which, if done in person, would normally be considered troublesome, such as compulsive gambling, or shopping, are sometimes called net compulsions.[4] Other habits such as reading, playing computer games, or watching a staggering amount of internet videos or movies are all troubling only to the extent that these activities interfere with normal life. Supporters of disorder classification often divide IAD into subtypes by activity, such as excessive, overwhelming, or inappropriate pornography use,[5] gaming,[6] online social networking, blogging,[7] email,[8] or Internet shopping.[9] Opponents note that compulsive behaviors may not themselves be addictive.

[...]

Over the past decade, the concept of Internet addiction has grown in terms of acceptance as a legitimate clinical disorder often requiring treatment.[11] However, known academic authorities take stances in either supporting or opposing the existence of Internet addiction disorder (IAD). In 2006, the American Medical Association declined to recommend to the American Psychiatric Association that they include IAD as a formal diagnosis in DSM-V,[12] and recommended further study of "video game overuse."[13]

Whether to call all this an actual “addiction” — or not — may matter only to one’s health insurance provider. If it were in the DSM, then insurance companies would likely have to pay for addiction “treatments”.

Mind you, this is a close fit, not a direct hit, but I think you could use the word technophile.

NOAD defines technophile as "a person who is enthusiastic about new technology."

I'm quite aware that enthusiasm doesn't necessarily equate to addiction, and that someone can indeed be a technophile while only spending a few minutes each day on a cell phone. (The word could perhaps more accurately be applied to someone who spends a moderate amount of time using mobile technologies, yet feels an almost constant urge to upgrade to the latest gadgets.)

That said, in context, I think something like:

Paul must spend nine hours a day texting; he's quite the technophile, don't you think?

would get an affirmative response from me, as opposed to a pedantic correction. After all, in the larger schemes of things, the technologies you allude to are relatively new. How new? New enough that my spell checker is putting red lines under both technophile and texting: