I've used the line a few times the last few years - generally daytime during the week, and my impression has been that most people travelling on it are locals - and trains seem to be reasonably well loaded. I have a friend who lives in Morecambe and would use the train rather than the bus to Lancaster because she claimed the train was cheaper - I don't know if that's still the case.
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Interesting point. I'd expect it to make some difference, but not a huge amount. It will certainly cause some reduction in traffic at the Lancaster end of the A683 and over the bridges. A long time ago at one of the council open events, it was mentioned that the aim was to take advantage of this to convert one of the 3 lanes on Greyhound Bridge to be a dedicated bus lane - I don't know if that's still the intention, but obviously if that does happen, that could help buses quite a bit.
But on the other hand, the by-pass will only be useful for traffic heading to/from Morecambe from the M6 or from the Caton direction - anything coming from Lancaster itself (which I would imagine accounts for most traffic) wouldn't use it. Also, the very existence of the by-pass is almost certain to attract many more cars, and that could easily lead to existing roads being more, rather than less, congested between Lancaster and Morecambe college (where the by-pass joins the existing network) and Morecambe.