How did dog get from Bedford?.

By train perhaps? A friend of mine had a guide dog once who boarded a tube train without him and left him stranded on the platform. He did not blame the dog. He was eventually deemed an unsuitable owner.

Bedford trains don't go to Brum though, JessM He'd have had to change somewhere along the way. He could be from the Olney end of MK though, they have 01234 phone numbers and then he could have got the train from MK to New Street. (London Midland or Virgin)
Perhaps, more prosaically, the owners had been away over the BH weekend and he had got lost then?
I would have said Bedford was more like 70+ miles form Sutton Coldfield, myself. I know it's 75 from us to Moseley and that's further south. Of course he could have hitched a lift on the M1. Who knows?

Stowed away on the back of a lorry, just when the lorry driver was distracted by the discovery of illegal immigrants on board? (only spotted this once, I hasten to add)

Thinking about it I think the dog must have jumped on the back of a lorry as the owner told me he was lost in a sub district called Cardington.
I used to live in St Albans but I have not heard of Cardington.
I think it is unlikely he would have come by train as it would appear that he would have had to have to have got at least 2 trains to Birmingham New St and then another train to Sutton Coldfield.
I think the dog would have been spotted if he had done all those changes and it would have had to have changed platforms at Birmingham New Street and I think it is unlikely the dog would have paid its fare.
I am not sure what a lorry would be delivering from Bedford to Sutton Coldfield.
Regardless I think the person who put him over the fence in to my back garden was very cruel as there could have been serious consequences.

Frank

Well, Frank, surely better than just leaving him to roam around lost?

A happy ending anyway and here's another happy ending

My parents had a border terrier X full of character. When they moved us all from Carlisle to Wolverhampton the dog decided to return to Carlisle. He disappeared for a week and it was the original neighbours who found him. He was transported from boundary to boundary and fed along the way by policemen. He never did stop running away and could squeeze through the tiniest hole but at least he learnt not to go quite so far!

The person who put him over your fence probably thought he lived there. If he could hear or see Andie he was probably acting as if he wanted to get into your garden. At least he was then safely with a doggy person.

Cardington is a very famous area, south of Bedford Frank - I am surprised you never heard of it. Famous for what you may wonder. Air Ship Hangars is the answer. Listed buildings I believe and I think used by the Building Research Establishment for some of their practical trials. (inside the Air Ship Hangars not on them - what would be the point in doing research on Air Ship Hangars unless of course you were planning to bring back air ships) .
It is also not that far from the big Amazon depot so perhaps the dog managed to get inside an Amazon delivery lorry - they ship to all corners of the realm, I am reliably informed.

Thank you Jess for sharing these fascinating facts with us

Sometimes dogs are stolen and then discarded by the thieves when they realise they've been identichipped [happened to my friends two labradors].