I'm pretty sure that I read recently that it's only required above FL130 in the UK, but of course the general recommendation is to use it abouve FL100.
Art. 92 applies so you can fly up to 30 minutes between FL100-130 without (supplemental) oxygen.
There are plenty of aircraft within the NPPL restrictions that can reach those altitudes. A Vans RV-7 recently flew above 26,000' without turbocharging. Above 18,000' masks rather than cannulas are recommended and that increases the flow rates considerably.
VFR in Class C is permitted with a clearance of course, but it might be advisable to make some prior arrangements. Also I'm not sure that your altitude encoder needs to be certified to the planned altitude. Many of them are only tested to 20,000'.