Hello Bob, and thank you for your question. I would be happy to assist.
It sounds like both you and your neighbour have legal rights to use the alleyway in question. (It is not uncommon for the benefit of such a right to be missing from the title to the land that benefits from it. The important feature is that it is detailed on the title which has the burden of the right of way - in this case, the title to the alleyway.)
The precise nature of those rights will very much depend upon the exact wording of the relevant title documents, but generally speaking, a right to use an accessway does not entitle a party to obstruct it - far from it. In fact, I would be surprised if there were not some wording in the title documents to expressly prohibit such action. In any event, where a right of access that one is entitled to is obstructed by a third party, one is perfectly entitled to insist that the obstruction is removed.
I note that you have emailed your neighbour on multiple occasions asking that the items be removed, without success. I would therefore suggest sending a formal letter before action, repeating the request that the items be removed and specifying a reasonable period of time within which this should be done. The letter should also make clear that, if the items are not moved by your neighbour within that timeframe, you will move these yourself. (In order to ensure that the letter is firm but objective you may choose to have it prepared on your behalf by a local solicitor - not least as the formality of a solicitor's letter is often sufficient to 'shock' the recipient into taking the necessary action. This should not be an expensive exercise as it should take only an hour or so of their time at most.)
If the obstruction remains in situ following receipt of the letter and expiry of the specified time period, you may then move the items. You should, of course, take care not to damage these, and to place them all within your neighbour's own property. (As tempting as it may be to handle these roughly or to simply dispose of them, you obviously do not wish to be on the receiving end of a claim for damages to the items! You should therefore take great care when dealing with them.)
If the obstruction reappears, it is possible to pursue legal action in order to enforce your easement, by seeking an injunction and/or damages. Such a process can be lengthy and costly, however, and I would therefore recommend that you do endeavour to resolve matters by agreement if this is at all possible, in order to avoid unnecessary cost, hostility and stress.
With regard to the rubbish in your neighbour's garden, you obviously cannot access his property to deal with these yourself. You may, however, wish to contact the Environmental Health Department of your local council, particularly if the nature of the waste gives you cause for concern with regard to rodents, pests etc.
I hope that this assists. If my comments have been helpful in clarifying the legal position I would be grateful if you would confirm by accepting and rating this response. This can be done by selecting one of the five stars located on the top right of the question page and enables me to gain credit for the time taken in responding to your query. This will not close the question and you will still be able to comment below with any additional questions or comments.