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It depends on the company. Usually if they call the scheme to £1000, usually they own it. If the cap is higher, usually the operator owns it.
See https://help.cyclescheme.co.uk/article/45-who-actually-owns-the-cycle-to-work-package for more.
If you do resign not sure you own it. I did resign once, paid the remaining without the savings, but at the end of the year, Cyclescheme asked me if I wanted to keep the bike or extend the lease until the end of 5 years.
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Substance over form, you own the bike after the 12 months period, put it this way, this is a scheme which really should be a personal allowance for individuals to purchase a bike BUT the powers to be decided that's too easy...
Why the heck does the employer and a 3rd party which takes up to 15% commission for doing fuck all is needed to buy a bike?
Yes and no, technically the scheme operator owns it.
That said, IF you do resign before the full instalment payment, you'll be charge the outstanding amount (wihtout the tax savings) as a benefit in kind and you'll then own it.