They had done that all through the 20s and the 30s, because the Finnish Army was designed and built for a big part by former German officers of the Reichswehr and a major contingent of the Finnish operational leadership had been trained in Germany (this includes, but is not limited to, the "Jäger" cadre). The co-operation between Finland and Germany was extensive. I guess my point is that there was quality training and experience present and already used in Finland, adapted to local conditions. Ten or even five years before British help would have certainly had a major influence, but I am at a loss to see anything major the Finnish military leadership could have learned from the British at this point. Anyway, most things we see as the British forte, such as naval warfare, did not necessarily apply to the Finnish situation at hand.
So, because the Finns have had technical support from a large nation in the history they'll
never need any support whatsoever.
By following that logic into the ridiculous the Swiss for example IRL wouldn't have needed any British training whatsoever regarding their Hawker Hunters bought in the fifties from the British. After all, the Swiss know what works best in their environment and have experience with that.
No offense to the Finns, but they're a small nation with a limited military. What experiences do they have with tanks and groundsupport for example?
I doubt there isn't much the Finns can learn with regards to new technical advances, use of heavy artillery, new modern doctrines about cooperation between air and land etc etc.
There were plenty of technological advantages during the '30s and '40s and new doctrines developed of which the Finnish army wouldn't have any experience with.
If anything they could use training with the equipment the Allies would supply them with.
Do agree about not much learning at that point.
Like I wrote earlier, I think it is quite unlikely Stalin would go as far as wage a protracted war against the Western Allies over Finland. Finland was a minor concern that could be taken care of later, not enough to warrant war with Britain. IMO the Anglo-French intervention would have brought the war to a halt before any allied contingent, save aerial, could have time to make it to the front at Karelia. What you are discussing here is interesting, but verging on the ASB territory.
I'd sooner expect the Allies to blink then Stalin. Or at least I think it's more likely for the Soviets to take their chances and go on with their Finnish adventure, then it is for the Allies to start supporting the Finns actively.
I wouldn't be too surprised if the Brits would tell the Finns how to run their army. They would feel superior to the little guys.
Maybe there are still some other stereotypes we haven't used yet?

Why would they?
We'd compare the Soviet support to Republican Spain during the civil war with this situation. There the Soviets managed to dominate the Republican politics and military thanks to their 'support' (dearly bought by the Spanish gold reserves). But during the Civil war the Soviets were the only nation supporting the Republicans and this gave them more influence.
In this timeline, there's also French and other support, so I doubt even if the British wanted, they would get any kind of control over the Finns or their army.