First of all, NHS treatment is not always actually provided by NHS staff. In recent years, NHS money has sometimes been used to fund alcohol treatment which is provided by charities or other agencies. These services are still free. In some areas one of these charities will provide all alcohol treatment, in others they'll provide some of it 'in partnership' with a local NHS trust. The service usually includes hospital detox, home detox, individual and group psychological help, but this varies around the country.
Usually you can be referred by your GP, another health professional, or you can refer yourself. Your local contact details should be easy to find on the internet. Each area will have its own way of providing treatment, so it's impossible to tell you here exactly what will happen.
NHS services are free, and they must be based on solid, sensible, evidence-based treatments. Many of their staff are very skilled and experienced and will do a superb job of helping you. They are usually very well connected to lots of other free services in the area, and can often point you in the right direction for help with accommodation, housing, employment, mental health, support with childcare and lots of other things.
In many areas, treatment places are limited. The demand for alcohol services is huge, and it can be that the poor staff are like a B road trying to cope with motorway traffic. So you may have to wait for an assessment, or once you've been assessed you may wait for the treatment you need. Some services maximise the number of people they can see by only offering group sessions, with no individual counselling, or they may limit the number of sessions you're entitled to. Alcohol services are sometimes provided by the same team, in the same building, that provide treatment for heroin and cocaine users. Some people find that this makes the waiting room an uncomfortable place for them to sit.
Always try your local NHS service first. Don't be tempted to pay for expensive private treatment until you've done this. Don't be afraid to be a little bit pushy if you're made to wait for a service. And if you're not happy with the member of staff you're seeing, you can, for whatever reason, request a change. Please always treat the staff courteously and respectfully, even if you're frustrated by the limitations of what's on offer. And if you think you've had a good service from them, tell them.