The clues as to time of day and weather in Chapter 1 of The Hounds of the Baskervilles are scant, but we can apply Sherlock Holmes's methods, as Watson does in this chapter, and make a few deductions of our own. Time of day is relatively easy. Watson, speaking as the narrator, says that Holmes rises in late morning as opposed to early morning and that, as Chapter 1 opens, he is seated at the breakfast table. Also, Watson is just coming out to breakfast himself because Holmes informs him that the coffee pot is front of him. Had Watson already breakfasted, he would not need to be told. You only have to determine what is generally referred to as late morning and still breakfast time as opposed to an early lunch time to know the time of day.
As to weather, Holmes looks out the window from where he is seated and can see a man and dog approaching from the other side of the street (from a seated position in a chair, he couldn't see anything nearer at hand). By the time Holmes gets to the window to look out directly, the man and dog are at the doorstep and the visitor proceeds to ring the bell. From this, you can deduce that there is no London fog, nor any rain nor snow. The day is fair with sunlight enough to fully illumine an approaching visitor; so if there are clouds, they are not heavy, darkening clouds. Further, when the visitor, Mr. Mortimer, enters, he is wearing a frock-coat but no top coat as he would do in cold weather. Additionally, frock-coats were worn year round, so unless the fabric were mentioned, and it isn't, it is harder to deduce weather as related to seasons. Watson does say the frock-coat is dingy and the trousers frayed. This may bespeak weathering the rainy season, which in England may be autumn. From these clues, you can pinpoint a reasonable estimate of the weather in Chapter 1 of The Hound of the Baskervilles.