The actress shown interacting with young Tolkien is Genevieve O'Reilly, presumably during a break while young Tolkien is playing rugby. Interestingly enough, the character she plays isn't named, even on the imdb site (where the actress's name is given but the character name left blank).*
My guess? Aunt Jane.
Though that's just a guess, and it cd just as easily be some reimagined version of Jenny Grove (Edith's cousin and later companion) or that favorite easy-out of 'based on a true story' biopics: composite (i.e. fictitious) character.
Here's another link (like the one above, taken from the Tolkien Society's site), showing some on-location setting (this time in Cheshire) said to feature in the film:
One significant bit of casting since I last posted on the topic is Colm Meaney (Mr. O'Brien, Transporter Chief on STAR TREK: THE NEXT GENERATION) as Father Francis. I wonder if this means they're making Fr. Francis Irish or if Meaney will be assuming a Welsh/Spanish accent.
Another interesting casting annoucement, for reasons having to do with the character rather than the actor, is the addition of Sam to the list of characters -- no, not the hobbit Samwise Gamgee but apparently Tolkien's batman (aide) during the war.
* There's now a wikipedia page for the film, which also omits the character name https://en.wikipedia.org/wiki/Tolkien_(film)