The company dates itself back to 1905 when the current owners Grandfather first started work as a luthier in Madrid, Spain and there is a very full history of the company and the family on the website. From reading the history it is highly likely that the previous generations of Rodriguez made some Ukuleles but as there were not labelled it would be very difficult to prove. 

On to Manuel Rodriguez III and the current set up. The firm has grown from a single luthier to a full blown factory and is in Esquivias, (between Madrid and Toledo, Spain), They have been one of Spain's biggest Guitar makers for a while now, making acoustic classical and flamenco Guitars and in 2013 they introduced a couple of ranges of Soprano, Concert and Tenor Ukuleles to their catalogue using a variety of woods. In 2015 they started branding imported Chinese made models, I have not seen them on the website but I have seen them for sale in the U.S.

At some point after 2015 Gewa took over the firm and both Ukuleles and Manuel Rodriguez III were removed! I don't know the full story but I don't think it was particularly amicable and at one point Manuel Rodriguez III was looking to set up a rival company, (possibly in Fiji?). From the current website it doesn't look like there has been any reconciliation?