.....does it makes sense to get the train or bus tickets online ahead of time or is it really easy to get it there?...
For the bus - yes, at least a day or two ahead of time. Easy to do if staying in the area and can walk by the bus stations. Buses will fill up. Trains never because if a seat is not available, you stand till one is. Lots of people travel between the airports and London with luggage. There are some small luggage storage that can be used or keep it by your feet -- NOT IN THE SEAT !!!
....And as far as times, if we get the tickets online for a specific time but end up being a little late, is that ok?.....
No, if you ticket specifies a certain train and time, then it is only good for that train at that time. However, if you are actually asking about the subway, Tube, or the "trains" to and from the airport, then it is an open ticket and makes no difference when you use it. Just put the ticket through the turnstile (mag reader) and you are good to go.