I'm currently using a Shimano Tiagra road mech

One of those

I use it with a 32 tooth chainring and an SLX cassette (28/11)

I will probably need to renew my cogs soon and I was wondering if the mech will cope with a 32 tooth sprocket at the back or do they only work with smaller ones (i.e. 28) like Im running now?

Any advice appreciated

Yep , 28 is your max. That's the set up I've been running for a couple of years. Saves a fortune and less mech ripping!

Might be worth borrowing a wheel with a bigger cassette on to see if it fits. I've used a road mech with an 11-34 before now without a problem, just depends on the dropout thickness and distance between the mech mount and the axle on your particular frame.

I've used a short cage XTR mech (only a fraction longer than your road mech) on a 12-34 cassette before without a problem.
It should work fine but you'll need to adjust the b-tension screw on the back of the mech.

road mechs are officially okay up to 27t sprockets. In reality 28t is also fine, and sometimes more – depending on dropouts/chain tension etc.

It's not the length of the cage that's the reason though – a common misconception – but the geometry of the mech. For example long cage road mechs are also only rated to 27t officially.

You'll be fine 59. If you want to run a 11-32 cassette though then you'll need a mtb mech.