Being caught urinating in public could put you on the sex offenders register.
Very unlikely. Being convicted of exposure requires an intention to be seen by others, and not everyone convicted of that offence has to register.
"Whilst at medical school, a Dean pointed out to us that anybody caught urinating in public may end up on the sex offenders register"—via email, 23 June 2015
We received an email recently asking about the number of doctors that are sexual offenders (our answer to that is here) mentioning this claim, which seems to be of general interest. Our response to the Dean is that it's very unlikely.
People have to register with the police after being convicted or cautioned for certain, listed sexual offences. The list includes the crime of "exposure" of the genitals, but to be convicted of that the person must have intended to be seen by someone and to cause them "alarm or distress".
The Scottish offence of "sexual exposure", which also gives rise to registration, specifies that the exposure has to be "in a sexual manner", as well as intentional.
A clandestine duck behind a wall isn't likely to satisfy these kinds of criteria, although a very public display is obviously more risky.
Urinating in public without the intention to expose oneself and "alarm" others will usually be charged under a local byelaw or the Public Order Act 1986, according to the College of Policing. That kind of conviction is not "sexual" and doesn't require registration as a sexual offender.
Even if a court felt that there was a sexual element to your behaviour, the requirement to register as a sexual offender doesn't follow automatically. Unless someone who saw you is under 18, you wouldn't have to register if fined rather than sent to prison.