In terms of ingredients you can get ones that are vegan.  Careful as some tablets contain lactose - however this is an insouble sugar and so soluble tablets/effervescent tablets are less likely to contain lactose.
Unfortunately I can't say which brands of paracetamol contain no lactose as it would vary from company to company.  Bear in mind that many drugs only list the active ingredients on the packaging and so you may need to check the PIL (patient information leaflet) for full lists of ingredients.
Steer clear of capsules - they will generally contain gelatine.  Some liquid preparations may be ok, but their colourings may not be vegan - again this would vary from company to company.

That's the ingredients....however pretty much every drug available has at some point been tested on animals so would not be vegan from that point of view.  Many vegans will take medication as it is seen as something that can't really be avoided.  Others won't take drugs such as analgesics - this is really up to you.  In some circumstances I will take medication, but if I can avoid it I will.
I don't think that you should have to suffer the pain and in your shoes I probably would take painkillers, but everyone will be different.

You could try visiting a homeopath - I don't know what homeopathic medicines to recommend for pain - I am a pharmaicst who only deals with conventional drugs.  I tried to have a look to see if there was any natural pharmacies in Rotherham but unfortunately couldn't find anything (admittedly I only did a quick google).  Some local independent pharmacies will specialise in 'natural' remedies and it may be worthwhile phoning round a few to see if any have this speciality.

Hope this has been of some help.

eta: the reason I recommend trying to find a natural pharmacy rather than seeing a specialist herbalist is that you can go and chat to the pharmacist for free whereas herbalists normally charge a consultation fee (plus I am biased being a pharmacist myself ::))