I am in HR and I do get cross when people don't take all their leave (mainly cross at the managers for not having managed it over the year) and then asking to carry over.
We do allow 5 days but we give 5 more than the statutory minimum so we can.
Statutory minimum (which is 5.6 weeks, or 28 days if you are full time) should be taken in the holiday year. It is technically illegal for an employer to pay it out (as that encourages people not to take it and the point of it is that people take it as they need a rest) and same re carrying over - it should all be taken in the leave year.
An employee has a right to complain if an employer has repeatedly refused them requested leave (and the only reason an employer can refuse is 'operational reasons', but that covers pretty much anything) but that right fades away when the employee has never requested it, or not requested it in good time (though there is some case law on that I can't find at the moment).
The employer has a right to set a holiday year, which employee should be informed of. They also have a right to set time periods by which holiday should be requested and to limit the amount taken at once, as well as to force holiday to be taken (for example, many employers shut over the Christmas period and force employees to take those days from their leave entitlement).
www.gov.uk/holiday-entitlement-rights/booking-time-off-
There is no compulsion on employers to allow any carry-forward.
I have worked for employers who hoped employees would not request their leave and then have the year run out on them. Most are not like that.
I usually end up allowing people to carry forward more (mainly if they give me the dates they will take it though!) and sometimes I agree to pay some out (though never to take them below the statutory minimum for the year, bearing in mind they may have carried forward from the year before too) - in this case I would certainly be OK if they said 'look, I know I should have booked sooner, but what's the next available dates I could take which don't cause any problems - I'll take those' - I'd agree to that.