I.
It is proven from the Prophet (blessings and peace of Allah be upon him) in the authentic (sahih) hadiths that our blessings (salawat) upon him reach him.
It was narrated that `Abdullah ibn Mas`ud said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Allah has angels who go around on earth conveying to me the salams of my ummah.” An-Nasa’i, 1282; classed as authentic (sahih) by al-Albani.
It was narrated from Abu Hurayrah that the Messenger of Allah (blessings and peace of Allah be upon him) said: “There is no one who sends salam to me but Allah will restore my soul to me so that I can return the salam to him.” Narrated by Abu Dawud, 2041; al-‘Iraqi classed its isnad as good (jayyid) in Takhrij al-Ihya’, 2/764. Shaykh al-Albani classed it a sound (hasan) in Sahih Abi Dawud, 1779.
Whatever is soundly narrated from him (blessings and peace of Allah be upon him), we are obliged to accept it as true and believe in it, whether we know how that thing happens or not.
II.
The conditions of al-barzakh (the period between death and the resurrection) are matters of the unseen; we cannot comprehend their reality or nature, so we believe in them as mentioned in the texts, without enquiring into how they are.
Ibn Hajar (may Allah have mercy on him) said:
With regard to what Abu Dawud narrated via another chain of narrators from Abu Hurayrah, attributing it to the Prophet (blessings and peace of Allah be upon him): “There is no one who sends salam to me but Allah will restore my soul to me so that I can return the salam to him,” and its narrators are trustworthy…
Some people may find it problematic from another angle, which is that this would take all the time, without a break, for him to return the salams , because the salams will be coming to him continuously from people all over the world, who are too many to count.
The response to that is: matters of the hereafter cannot be comprehended by (human) reason, and the conditions of al-barzakh are like those of the hereafter. And Allah knows best.
Fat-h al-Bari by Ibn Hajar, 6/488.
Matters in the modern world can give us an idea of how that may be: through modern means of communication, a man can send a message and that message could reach millions of people. So the idea that the Prophet (blessings and peace of Allah be upon him) can return the salams of millions of people is not something that is beyond the power of Allah.
See also the answer to question no. 34561.
And Allah knows best.
