How to get a new UK Driver's licence while travelling?

If you are a UK tourist in Europe, the EU site also has some information - e.g. lost/stolen licence: contact the local police and then talk to the embassy/consulate (for the UK, they'll just refer you to the DVLA). http://europa.eu/youreurope/citizens/vehicles/driving-licence/driving-licence-loss-theft-replacement/index_en.htm

If your licence was stolen, you'll need a Police report (reference/copy) regardless of where it occurred (UK or abroad).

Please note the DVLA stopped issuing paper counterpart licences on 8th June 2015. If you've got your licence number, postcode and national insurance number to hand/noted down somewhere (you need all 3), you can use this to retrieve your licence information as a digital or a paper copy using the "View or share licence details" online tool on the .gov.uk site.

Follow the links at https://www.gov.uk/browse/driving/driving-licences

You can usually use the online process on the .gov.uk site to apply for a new/replacement licence (use the interactive tool in the answer above, the link redirects to the new site) or by phone.

The number below can be called from abroad by substituting the first 0 with +44, e.g. 0044 300 790 6801

You can contact DVLA to apply by phone if none of your details have changed.
DVLA driver licensing enquiries
Telephone: 0300 790 6801
Monday to Friday, 8am to 7pm
Saturday, 8am to 2pm

also, from the .gov.uk Driving abroad page - just to emphasise there are no immediate changes to the processes involved when travelling in the EU:

There has been no change to the rights and status of EU nationals in the UK, and UK nationals in the EU, as a result of the referendum.