You'll be fine. Official guidelines as far as I can tell are that tattoos are tolerated as long as they aren't offensive (I.e. avoid swear words and swastikas.) I also have tattoos on my forearms, and I am very likely to get more. And I have 16 above the neck piercings, and 14mm tunnels in my ears. I tend to hide them for getting through interviews as interviewers are likely to be conservative, but in terms of being on placement they can't chuck you out for having tattoos or piercings (as long as you remove jewellery when appropriate.)
Bear in mind that the conservative elderly are not the only group doctors have a duty of care to. I think there is already an issue with the vast majority of doctors being very privileged, well off people. This might work for your granny, but it's likely to alternate huge numbers of (for example) young, LGBT+ young people presenting with mental health issues who need to feel that their doctors are accessible, and are at least marginally able to comprehend their lives.
I'm also not even sure it does work for your granny. My granny never stops complaining about how doctors are all rich, private schooled men who have never smoked a cigarette in their lives and can't possibly imagine what it's like to be her. (Obviously, this is an exaggeration but these feelings are symptomatic of a genuine problem.) The very narrow definition of "professionalism" that we are expected to comply to alienates as many people as it reassures.
Also bear in mind that until very recently, anyone who wasn't white or male was also considered an inappropriate kind of person for medicine. Tattoos and piercings aren't the same as being a woman or a person of color, but the reason they are considered unprofessional is due to strong historical associations between tattoos and the working classes, and between tattoos and radical, LGBT+, activist communities. The implication is still that there is a certain kind of person that should be doing medicine, and its not the kind of person that ever prioritised something else over "professionalism."
We need a diversity of people within medicine, including those of us who weren't planning to be a doctor at 13 and living accordingly. When I graduate, I feel like I'll be the only doctor able to advise a teenager how to look after an infected scaffold piercing (which we don't get taught in med school, BTW), and one of the few able to relate to homeless LGBT+ youth. For my career aspirations, putting these people at ease and helping them feel like their doctor is someone they can talk to, is a priority. Someone else can work in Tory geriatrics.
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