The RAF general guidance (from Dress Regs) is:
Tattooing – General Guidance (AP3392 Vol 5)
0158. The Royal Air Force require personnel to present a high standard of appearance when on and off duty irrespective of the fashions prevailing amongst the civilian population. Tattoos and henna style markings should not to be visible when on parade in No 1 SD, if they are not covered by the uniform they are to be concealed cosmetically or by adhesive plaster.
0159. Tattooing is to be discouraged, and unacceptable tattoos or henna markings, whether ordinarily visible or not are not permitted. Such tattoos are defined as being any tattoo(s), excessive in size or number, obscene or offensive (e.g. racist, profane, blasphemous, anti-Semitic, overtly sexist, homophobic, drugs-related or of an extreme political nature).
0160. Detailed rulings and administrative instructions on tattooing are contained in AP 3392 Vol 5 which is the definitive document for the RAF policy on tattooing.
And the detailed stuff is extensive and mostly concerns action to be taken, but the intro to the regulation gives the background and justification:
Introduction
1. Tattoos are a popular fashion decoration for male and female personnel and are generally a personal matter. However, the RAF, as a disciplined Service, requires higher standards of appearance and bearing than might be acceptable in other walks of life. Self-respect is an important part of the core values and standards of the Service; it indicates dignity, integrity and professionalism. The nature and position of an individual’s tattoos cannot therefore be ignored when forming a judgement of that person’s character. Likewise, the impact of tattoos on other personnel, for example if they are offensive, is an important consideration in deciding whether or not a tattoo is acceptable. The policy applies to those individuals with pre-existing tattoos seeking to joint the Service and those personnel who acquire tattoos whilst in the RAF; therefore, when reading the policy, where reference is made to commanders, it should also be interpreted as applying to recruiting and selection officers.