General SLT podcasts which cover a range of topics:

See Hear Speak (Tiffany Hogan) - interesting conversations with people who care about reading, language, and speech in the developing child
Speak Up: A Speech Pathology Australia Podcast - 50 podcasts on different topic areas
The Speech Link: A Speech Therapy Podcast with Char Boshart (SpeechTherapyPD.com) aimed at those working with school-age children
SLP happy hour - 'for caring SLPs who want more clam and less chaos in their workday... we talk about happiness, SLP lesson fails, easy (lazy) lessons, new habits we are trying to create and tips and tricks for maintaining more SLP calm.'
Speech science (MWH Production) hosted by Matt Hott, Michael McLeoud + Michelle Wintering
Teach me to talk - Helping parents teach toddlers to understand and use language
The working therapist (Haden Boliek) 'focussed on providing quality speech [and language] therapy'
The SLP Now Podcast (SpeechTherapyPD.com) - 'your go-to resource for practical speech-language pathology tips.
Conversations in Speech Pathology (CSP)
The Informed SLP - aims to connect clinicians with researchers
Talking with Tech

Aphasia

Inside Health (R4) Stroke man recovers speech After a stroke Peter lost the ability to speak. For four years he wrote notes to communicate. But after a second stroke, he could suddenly talk again. He tells his remarkable story (28 mins, but includes other health stories). More about Peter's story here

Developmental Language Disorder (DLD)

The Life Scientific (R4) Dorothy Bishop on Language Disorders (30 mins)

The Brain

The Infinite Monkey Cage (R4) The Human Brain (30 mins)

In Our Time (R4) The Brain (45 mins)

Mind the Gap (R4) A History of the Brain (15 mins)

Radiolab When Brains Attack (60 mins) Strange stories of brains that lead their owners astray, knock them off balance, and, sometimes, propel them to do amazing things.

Dementia

Word of Mouth (R4) Communication and dementia (30 mins)

AAC

Talking with Tech AAC Podcast (Rachel Madel and Chris Bugaj)

Autistic Spectrum Disorder (ASD)

Word of Mouth (R4) Autism and communication (30 mins)

The Life Scientific (R4) Francesca Happe on autism (30 mins)

Conversation

Word of Mouth (R4) 'Real Talk' Conversation analysis (30 mins)

Word of Mouth (R4) Small Talk (30 mins)

Word of Mouth (R4) Greetings (30 mins)

Word of Mouth (R4) The power of telling stories (30 mins)

Linguistics

Lingthusiasm (Gretchen McCulloch & Lauren Gawne) 'A podcast that's enthusiastic about linguistics'

The Language Doctor (WS) Idioms (13 mins)

The Language Doctor (WS) Conditionals (13 mins)

The Language Doctor (WS) The Causative Have (14 mins)

The Language Doctor (WS) Phrasal Verbs (14 mins)

Dyslexia

Word of Mouth (R4) Dyslexia with Prof Maggie Snowling (30 mins)

Transgender

Word of Mouth (R4) Language and gender identity (30 mins)

Becoming Myself: Gender identity (R4) Trans men (28 mins)

Becoming Myself: Gender identity (R4) Trans women (28 mins)

Dysfluency

The Listening Project (R4) Wendy & Christine - Speech Therapy. Hiding her stammer was limiting her life, but now Wendy has found her voice. (4 mins)

Word of Mouth (R4) Stuttering and cluttering with Rosmarie Hayhow (not currently available)

Dysphagia

First Bite: Fed, fun, functional (SpeechTherapyPD.com)

DeafBlind

A sense of Texas

Disability

Word of Mouth (R4) Talking disability with Samantha Renke (30 mins)