All things being equal, 6 weeks is generally enough time to get a problem of that nature sorted out, with the time being between the first complaint and all works on the ground completed.
For me if things take longer than normal its usually because after serving notice on the landowner they then prevaricate and problems take MUCH longer: -
e.g. ask them nicely to sort out so give them ~7 days (generally), then if nothing is done serve notice requiring them to open it, so thats another 2-4 weeks depending on caseloads / etc, then at the last minute they say 'well I'm only a tenant so write to Mr X up the road' (despite them saying they're the landowner), so that's another 2-4 weeks, then the landowner says 'ok I dispute this notice because I have evidence the route isn't a BW / the route is wrong / whatever'. So, it may appear a simple obstruction but occasionally its far more complicated.
I always do my best to update complainants if there are any problems like this though, so hopefully its just an oversight on your local teams behalf. Speak to them on the phone, its the best way to get an answer there and then
It's not time yet for a Section 130 notice, these are only useful in situation like towzers (which simply isn't good enough TBH)
Shout if anyone needs more info
Regarding the 'well they spent £x on path Y so they can't be short of money', part 2 of my - now ancient - article explains that the money for such work comes 99% of the time from central government capital grants; completely separate from general council maintenance budgets, despite them being used for broadly similar things and delivered by the same people.