There can be many reasons for anxiety or depression — nutrient deficiencies, malfunctioning brain biochemistry, adverse life-circumstances, social isolation, physical inactivity, deep-seated trauma, and so on. Meditation can do little for most of these. Where meditation is appropriate, the conclusion of metastudies is that it produces small to moderate improvements in psychological health. It's best used as an adjunct to conventional treatment (psychotherapy, prescription medicine) rather than as a treatment by itself.
Following the Beatles' visit to India in 1968, there was a lot of popular interest in mantra meditation. This lasted well into the 1970s. It turned out that the actual technique may not matter much. Jonathan Smith at Roosevelt University showed that sitting still with your eyes closed was just as effective as Transcendental Meditation.  The control group in his study practiced an invented form of meditation called "Periodic Somatic Inactivity." The instructions for PSI went like this:
Sit up straight in your chair. Place your feet flat on the floor. Place your arms in a position that is comfortable for you. If you are carrying a handbag or package, put it aside. Now, simply close your eyes and sit for the next 15 to 20 minutes. Remain physically inactive for this period of time; that is, sit still and avoid unnecessary movements. Let your mind do whatever it wants. Whatever you do mentally will have little or no impact on the effectiveness of the technique. The important thing is to remain physically inactive. Do not talk, walk around, or change chairs. You may engage in an occasional action such as shifting your position or making yourself more comfortable. And you may scratch. At the end of the session, open your eyes, breathe deeply a few times, and continue with your everyday activities.
PSI sounds a bit like shikantaza — literally "just sitting."
What's important is that you sustain your meditation practice every day over the long term. Over half of both Smith's TM group and his PSI group dropped out of the study. Either you must be internally motivated and disciplined, or you need to build an external support network of teachers and fellow meditators. It helps to choose a practice you enjoy and are naturally drawn to.
Certain kinds of meditation — Mindfulness Based Stress Reduction (MBSR) and Mindfulness Based Cognitive Therapy (MBCT) — are specifically designed to treat mood disorders.  Since this is their explicit purpose, it's reasonable to think these might have the best chances of success. In addition, they have been tested in controlled circumstances and the results published in peer-reviewed journals.
One problem in extracting meditation practices out of broader traditions is that you're assuming they are "detachable techniques."  The Buddha himself did not teach meditation; he taught the Noble Eightfold Path. Meditation was only one part. Similarly for Christian Meditation. The assumption is that you are a Christian first and a meditator second. The meditation is an optional way to express your relationship with Jesus. If all you do is practice meditation, you are missing something of vital importance.
The metastudies mentioned earlier report averages of averages. Individual results are going to vary from these averages. Some people do really well with meditation; others see no results at all or even increased anxiety. You can't tell in advance which category you'll fall into. The only way to be sure is to try it.
The controlled studies necessarily involve relatively modest amounts of meditation. You can find anecdotes where people have achieved much better results with much more practice. For example, there's a thread on Reddit, "How I defeated my 20 years of anxiety disorder with 5 hours meditation a day."  This, of course, is just one individual. There are no systematic studies of people who commit such huge amounts of time.
Here are some ways you can follow up on this post:
 Jonathan C. Smith, "Psychotherapeutic Effects of Transcendental Meditation with Controls for Expectation of Relief and Daily Sitting," Journal of Consulting and Clinical Psychology, vol. 44 no. 4 (1976), pp. 630-637. The actual techniques of TM are given at http://minet.org/www.trancenet.net/secrets/checking/index.shtml. TM-like practices are described in more detail in Joanie Higgs' Effortless Deep Meditation (Prominence Publishing, 2018) and in Yogani's Deep Meditation (CreateSpace, 2012).
 See for example Linda Lehrhaupt and Petra Meibert's Mindfulness-Based Stress Reduction (New World Library, 2017) and Zindel Segal, Mark Williams, and John Teasdale's Mindfulness-Based Cognitive Therapy for Depression (Guilford Press, 2018).
 Anna Lutkajtis, The Dark Side of Dharma (Aeon Books, 2021).