Really surprised by that... Where I've worked there's always been one HR woman (always a woman!) and she deals with the hiring, firing, disciplinaries, pay etc.... Maybe the roles are only split in larger companies?
Depends on the size of the Company.
Small organisations may have a generalist HR / Finance person who covers both HR practices and Payroll (or outsource the payroll and deal with the transactional / inputting side) and they are probably the Company secretary too or PA as there isn't a need for a HR person. They even be unqualified and have just picked up aspects of the role over time and do an ok job at it.
Medium size companies will likely have a payroll person, normally sits under finance but has links to HR.
Larger companies are likely to have Payroll Manager and 1 / 2 Payroll admin. It's likely to sit under 'Comp & Bens' under HR or Finance; and would deal with everything from payroll to loans, cycle to work schemes, pensions, pay reviews, pay scales etc etc. It's a huge area and is very specialist and boring so it pays well in my experience as there aren't many people who specialise in this area.
Payroll = numbers and a lot of detail, excel, inputting, tax info, pensions etc.
HR = development, strategy linking to people strategy and other HR plans like training, recruitment, performance of the business.
They are very different to me. I now have overall accountability of payroll (£7.5m per year salary bill) but I don't personally pay it, the payroll dept do. But I have to coordinate the right information to be sent to payroll if that makes sense.