...but that is completely unacceptable!
The flag of the red dragon, Y Ddraig Goch, is the flag of EVERY Welshman and Welshwoman, whatever their language(s) - Welsh speakers, non-Welsh speakers, learners, immigrants or whatever. And the Cross of St George is the flag of England and it's inhabitants - not English speakers. English is spoken in many countries, including Wales.
Some sites even use the Union Jack to represent English.
The Union Jack is the flag of the United Kingdom, which includes Wales (where some people speak Welsh), England (officially English only, but in practice many, many languages), Northern Ireland (English and Gaelic) and Scotland (English and Scots Gaelic) So completely wrong!
I believe that the only option is to use the name of the language, in the language, so 'Cymraeg' and 'English' (and perhaps 'Castellano' in South America and 'Español' in Europe, Български or whatever)
Here's an example of a site we developed for an EU-funded project, that involved people from six countries/regions: