Many of the trips I will take on the train involve one or more changes. The schedules often have 15 minutes between trains. Is this plenty of time? I am so used to slow airports that I am paranoid about changes and getting to the next plane/train.
At most stations that will be plenty of time as most stations are relatively small.
Trains are quite a lot different to airports. Train tracks are pretty narrow, really, and tend to be parallel to each other in a station, so even if you have to use the tunnel or bridge to cross across all the width of the station it doesn't take too long. Many of the stations which don't have lifts have ramps, but they can take a bit longer than the stairs because of all the extra distance.
No security checks, but some stations will check your ticket as you get to the way out.
Some of the larger stations have minimum connection times, like Birmingham New Street, Bristol Temple Meads, Reading and the such - just because of the time it takes to walk.
Even so I don't know of any longer than 15 minutes.
If you post your itinerary with changing points there are several folk here who can tell you if you are setting yourself up for any problems.
When you put in on www.nationalrail.co.uk for a journey, they know how long can be allowed to change trains at each station. Only those changes thought to be viable in terms of time are coming up on the search. In the event that you miss a connection due to a late running train - you have the right to catch the next train with no financial loss to yourself - even if it is a different company. If more than 1 hour late at your destination, you would have the right to claim compensation.
The trains generally run to time and often pause for just a couple of minutes at stations. Platforms are level with carriages.
You can count on most trains to run at least hourly. If a specific connection worries you, check ahead for what the next option will be. Advance (discount) tickets lock you into specific departures, but AnyTime fares don't. I understand you're in Britain now, so if you're buying tickets from a live agent, they could also offer opinions about specific stations or about which lines have less service to choose from. Do have your bags organized and be ready to hop off each train right upon arrival, to give yourself the maximum transfer time.