Actually, not many FTTC cabinets need planning permission - most go through an electronic submission (but still with a 1-month notice period) that Joe Public never finds out about.
Can you post a link to those plans? I'd love to see what detail they end up going to for it!
As for timescales...
BT's map, on its Infinity site, can get out-of-date (as do most marketing materials). BT Openreach is the best place for finding out their plans at an exchange-level: Openreach "where and when"
One good rule-of-thumb is for a cabinet becoming ready about 3 months from the first signs of earthworks - which is usually the appearance of a concrete base for the cabinet, complete with some ducts, and possible some new manhole covers placed outside. But, because most cabinets don't need planning permission, I don't think there's a rule-of-thumb that goes with PP. In fact, you'll probably create our first one
Once a cabinet is ready, it generally appears on the checkers with "proper" dates (rather than the quarterly "preliminary" dates) dated about 2 weeks later. Orders will be taken on the named date, and fulfilled around 2 weeks afterwards. However, the exchange doesn't usually go live until there's a good batch of cabinets ready (10 or so), so a very early cabinet may wait longer.
The thing most out-of-direct-control to BT seems to be the works to connect power. That has its own rules of planning/co-ordination with the council; that will take a good chunk of that 3-month wait to happen. If things go wrong with the power, then delays of 2 months can happen quite easily (usually becaue of rules for co-ordinating roadworks).
Another place for you to check progress is to check your council's highways website, and look for planned roadworks (tick any boxes to make sure you include works that are just on the pavement, if it has that kind of thing). That will help you see the plans for power - which was the only thing visible locally, not PP.