To be honest, I haven't heard any stories of people being let down on an airport trip by any local company; but you may know differently of course. The big ones like A1 (01709 555555) have hundreds of cars, so if one breaks down, or isn't available another 30 would be chomping at the bit to get the job.
I have zero connection to this company btw, just offering help.
I got the train once from Swinton to t'airport - found it quite easy, was only a weekend trip mind so not lots of bags. If I remember right they had lots of lazy walker things as it was a long trek to the terminal. Things might have changed since though it was 20 years ago!