Charity shops may accept towels if they are clean and in very good condition: not all do, so check first.
You could also try vets, animal charities and kennels. Local vets: Drove Vet Surgery (01672 512043), Drove Vet Hospital (01793 522483) or Riverside Vets (01672 514875).
Keep a few old towels at home for mopping up spills (great for plumbing accidents), kneeling on, turning into cleaning cloths or wrapping round pots in garden to protect from frost.
If they are not in reusable condition, tie them inside a plastic bag and mark ‘rag’ before placing in kerbside recycling black box or taking to Marlborough Household Recycling Centre.