They are part of the NHS, but they're unlikely to want to know an NHS or NI number.
nhs.uk/Service-Search/Sexual-health-information-and-support/se16/Results/161/-0.0580888763070107/51.4956550598145/734/0?distance=25
Guys Hospital is probably your nearest, Royal London is an alternative if you're near the Overground, and the Waldron at New Cross also has a walk-in GP service. Any of those is should be able to go 'yep, having one would be a good idea' and make some suggestions as to where.