It depends what part of the country you are in. Your local timber yard might be able to help.
Apart from that, you could take a trip to WH Smiths and buy (or browse) a couple of the woodcraft magazines available, dealing with carving, whittling and turning, and I'm sure there must be specialist timber suppliers advertising.
Or why not try the internet?
A Google search would doubtless produce a few hundred leads - be sure you include UK in your search conditions otherwise you'll get a few million leads.
For example with "lime wood source UK" I came up with Associated Timber Services, Grantham, Lincs (www.associatetimber.co.uk
) among many others. They might not supply the small quantities you need, but an email to them might get some advice.