No worries OP, in theory no graduate scheme these days, public or private sector will have an explicit age limit because of discrimination laws. In reality they do still tend to be majority recent graduates purely because most will have quite high expectations in terms of willingness to relocate around the country, work long hours, study on top of a full time job etc. But the major public sector ones at least will usually have at least a few 'mature' candidates each year and if you do have young children, or other caring responsibilities etc. allowances can usually be made to avoid disrupting them too much.
I disagree that all HR is all boring systems/compliance work as some PPs said, like all corporate jobs there's some of that of course (and recruitment is particularly heavy on it), and also typically most HR people in whatever specialisms experience some frustration at slow corporate/bureaucratic mechanisms, having to persuade/negotiate with business management to make improvements and implement HR/people strategies, having to work within limited funding and resources and with ever shifting executive priorities etc etc., but on the whole I'd say once you reach the middle/senior grades it is an interesting and rewarding job. I definitely think with your finance background and skills you would have a lot to offer esp in the reward space and possibly to workforce planning stuff too, I would say commercial and financial savvy is a common weakness in HR people, the number of relatively senior people I know who proclaim themselves 'not to do numbers' and can't drive a simple spreadsheet is a bit 
for the profession IMO.
The job is definitely not as 'people oriented' as people imagine HR to be though - by that I mean a lot of people have the idea we spend a lot of time directly talking to staff, delivering training, giving people bonuses and promotions and so on - some HR jobs are like that but for the majority normally we touch on individual staff only where things have gone badly wrong (grievances, disciplinaries, redundancies etc) or otherwise we work on more of a population/system level (when walking on reward schemes, learning and development etc). A lot of the 'nice' bits of people management like leading and motivating people, offering jobs/promotions etc, holding their hands and supporting them when things are tough (hopefully!) are done by line managers, and a lot of training these days is either elearning or delivered externally. So it's certainly not as 'fluffy' as people imagine it to be!
Re CIPD, I would certainly say most people in middle/senior HR roles in my experience have at least level 5, normally level 7 CIPD qualifications and/or a relevant masters degree. Nearly all MAs and MScs in HR Management will come with CIPD accreditation, but if you ended up wanting to specialise in OD for instance, some people go down the MSc in Occupational Psychology route so it might be better to wait until you are sure of specialism before fixing on a qualification.
I definitely think some kind of work experience in HR would be really useful for you, ideally getting to spend some time with different teams/specialisms as there's nothing like seeing it in action to understand the reality! Where I work we get quite a lot of requests like that from people in the business thinking about a career change to HR, and we always try and accommodate (and it doesn't make us think less of them for wanting a change from their current job although perhaps private sector is more cut-throat!) - as a minimum we give them some shadowing and sometimes we are able to offer some project work or similar to help build a CV (the odd CF does ask us to fund their CIPD which is a bit of a no as we always have a queue of people already working in HR for the limited number of funding spots available). Many people do get a bit disenchanted when the reality of HR doesn't match up to the idea in their head (as I say I think they see themselves dispensing tea and sympathy to struggling employees or spending all day delivering motivational speeches, neither of which enter much into any HR job I've had 
), but at least then they know if it's not for them before embarking on time consuming/expensive study or quitting their day job!