I've never seen it. I've dealt with the issue for three decades & never seen it as a primary cause. If the kid has a sleep disorder related to sleep apnea/tonsil size i could see it causing chronic health problems, but not changing the emergence of speech & language skills in a kid that was ready.
Tonsils. Enlarged tonsils alone are very common in young children. The tonsils don't usually cause delayed speech unless the tonsils get inflamed and cause ear infections. If the tube in the upper throat that vents the pressure in the ear gets blocked and infection in the ear causes the ear drum to perforate, the child may develop hearing loss which will affect speech development. A hearing test is key.