Erin,
We can go private whenever we like for mental health treatment like therapy. One of the drawbacks is that, aside from a couple of reserved job titles, it is an unregulated industry. You can take a 100 hour online course, never see a client and be a fully qualified therapist/counsellor in months and even join professional-looking associations (some are actually part of the online schools so make themselves appear more credible) to look like you know what you doing. The NHS tell us to be careful with this and look towards a handful of trusting registers like BACP who have a framework that includes minimum client treatment hours and ongoing supervision.
Getting into therapy, and certainly CBT, is as easy as a quick Google search and phone call. With private you might be in within days. But you have to do your research first.
With the NHS though you just see your GP or self refer (if you are under the IAPT framework which covers England). They can refer you to the Community Mental Health Team (CMHT). These guys do the outpatient stuff but they cover every form of mental health issue (they are multidisciplinary with nurses and psychiatrists as well as some community support roles) therefore you can expect a long wait in a queue with people who have not satisfied an inpatient need (Sectioning). Some CMHT's won't even touch anxiety due to funding. A 12 month wait to start treatment is likely (or longer). If you are under IAPT thought this system was brought in to sit between a GP level and CMHT level and provide faster access. About 30 days for Level 2 (self help levels based on CBT for maybe 5 sessions) but it may be more like 3 months for Level 3 (High Intensity CBT, potentially 10-15 sessions over a longer period with a higher trained professional). IAPT represents a CBT strip down though, which it has been criticised for, so it's a bit one size fits all compared to CMHT where they can combine therapies and for longer periods. IAPT people are mostly CBT trained.