I'm interested in taking up archery. Attending the closest club would require that I take the train there as I cannot drive.
Would I be able to take a bow and arrows, secured in a case, on the train?
The train operator is Northern Rail in the UK.
If the train operator allows it, yes, you would be permitted to do so.
There are no laws specifically against the carrying of bows and/or arrows, nor is a licence required for their purchase.
Bows can be carried openly with less issue, but arrows (because of their ease of use as a weapon) should be stored securely and safely, because
this protects you and others from accidental injury;
it prevents others using the arrow as a weapon;
it is affirmative evidence for a defence of lawful purpose, if you are accused of carrying an offensive weapon.
The latter is the greatest risk you may face from a legal perspective, so you should follow best practise similarly to firearms and ammunition: store and transport the bow and arrows safely, securely, separately.
In addition to the answer by @Nij, this would be like people taking potentially offensive weapons to and from martial arts events. I was taught that any offensive weapons should be carried in a bag, at the bottom of a separate bag. This was to make it hard (and shown to be hard) to access. I would do a similar thing with your arrows.
From my limited understanding, an offensive weapon is only such when it can be used, or looks like it can be used (e.g., replica weapons).