I'm planning a chest tattoo (after I've had top surgery) because the words mean a lot to me and having them on my chest will not affect my employment prospects.
That having been said, I seem to recall that you're in the West Country (ooh arr!) just like I am, so I'll give you a bit of local knowledge. I'm pretty high up in my field; I've been in management since the early 1990s.
- At my current employers, I have a colleague on our management team who has a wrist tattoo. It reads "Live your life". I'll leave you to figure out why she put those particular words in that particular location. She had it done in her lunch hours whilst working for us, and nobody was at all concerned about the fact that she has a tattoo (we were more worried about the reason behind it). She is client-facing so she represents our business to our multi-million pound clients.
- At my previous employers, I hired a man who is covered with tattoos. They can all be hidden by trousers & a long-sleeved shirt, but when he wears short sleeves you can see that he's carrying some serious ink. He wore long sleeves to his interview and we hired him for his skills and experience. When he later pitched up in biker leathers & a T-Shirt showing off his full sleeves, nobody batted an eyelid. Despite that, he did ask me if it was OK for him to show his tattoos in the office - he was worried that we might react negatively. I told him that he's a valued member of our team and his tatts are part of who he is so he should wear them with pride. He's been working there for about 9 years, and was with his previous employers for 14 years. However, it is important to point out that not all employers are as understanding.
- Many people in our neck of the woods have tattoos in visible places. Not many of them are doctors or lawyers, but a few are. To a certain extent, it does depend on what you plan to do for a living.
My advice? You already have a reasonably good idea of what you want to do career-wise. Think about whether you would be required to wear a suit & tie for that job. If so, a wrist tatt might be acceptable because it won't be visible. If you're not required to wear a suit & tie, the dress code is probably casual enough that a small, tasteful tatt wouldn't matter either.
But if it's big and would stand out, you might want to put it on the back burner until you're more established. Your wrist isn't going anywhere.