Remember that each MP's office works in a different way, to different schedules and communicates in different ways. So, the experience of other members may not match with your own.
Also remember that MPs are not obliged to help. Most of them do, either for your votes or to create the impression of being a good constituency MP or out of a sense of duty. But there is no legal obligation for them to intervene in the matters raised by you or by anybody else.
Most (not all) MPs offices list the way that they want to be contacted on the
Parliament website. Use the methods listed on the website as they are the ones preferred by their staff. Some MPs insist on receiving communication by post (snail mail) and many more (including my own constituency MP) will respond only by snail mail, even to emails. The reason for that is that parliamentary convention requires that they only deal with matters relating to constituents from their own constituencies and post is the one way of making sure that the constituents are physically located in their constituencies.
As an aside, I studied political science and one of the modules was MPs' communication with their constituents and the impact of social media. It also helped that some people in the course themselves worked in MPs' offices on a part-time or voluntary basis. Many, not all, MPs' staff are volunteers and not paid staff. Most also tend to be students, with typically the office manager or secretary being the only full-time person in the office. Hence the details in this post.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.