Be careful what you say in your last ditch attempt to get rubbish-owner to remove it. Those words could convey some kind of right that the solicitor could warn you against.
Also, whatever you do, don't move things onto your driveway or nearby, as they might be left there for another five years! With you being unable to get into your garage, plus liable (to council or neighbours) to have it removed!
And don't make any assumptions about what's been left in the garage; there could be noxious stuff, flammable stuff, etc; which could cost extra to be removed.
If you write to the rubbish-owner then (I'd say) tell them you will be "taking legal action", at their expense, if their items not removed by (date). Let them know you mean business, and cross your fingers.
Then use the time to read any policies that might contain legal coverage, so that when the time is up (or before) you can consult a bona fide solicitor and get that letter written. It should not only give a date for items to be removed, but also let them know that they will be charged for removal on that date plus storage/destruction charges. Obviously the solicitor will advise on what you can and can't do, and how best to manage the process.
You can decide at some point whether you just want the advice and that one letter, or more comprehensive assistance, from the solicitor.
Good Luck