Will a beech hedge grow in front of conifer hedge

By conifer, I assume you mean the dreaded Leylandii?
If the soil around it has not been worked for a long time the leylandii will probably have sucked all of the goodness out of it and I would not consider planting anything along side it.
Is this boundary the responsibility of your neighbour or you? This information will be available on your title deeds and the boundary which is your responsibility will be marked with and inverted 'T'.
There are few more posts which I have commented on regarding Leylandii Law and how the Council can help. If you cut the roots of their hedge and it dies you can be held liable for its death and they could sue. The neighbours are under no legal obligation to replace a fence along a boundary at all if it is indeed their boundary.

Thankyou so much. They have never touched the trees since planting. 30ft high now. The boundry is theirs. Their garden is higher than mine, what rights do i have if/when the fence blows down and all their garden junk and compost heap tumbles into my garden. Thankyou

It is not your responsibility to clear up the junk. Let them know of your concerns, vocally and if nothing is done, then in writing. If you have to clear it up there is nothing you can do apart from chuck it all back on to their side. If you have to pay to get it cleared you can take small claims court action against them for costs incurred. Take photos so you have evidence.
The hedge would be my primary concern.
You need to let them know that all evergreen hedges under the Anti Social Behaviour Act 2003, section 8 are to be maintained at no higher than 2 metres.
If they refuse to do this you will need to contact your local authority. They will charge you for the privilage of getting involved, but they will be able to get a court order to insist the culprits cut the hedge and maintain it in the future. Let your neighbours know that costs can be anything in the region of £1,800 if you get the council involved and your neighbours will be liable for that cost.
If you want any help writing your neighbours a letter, let me know. It sounds bizarre, but you should send it recorded delivery so you have evidence that they have received the letter.

They are very inconsiderate, speakers blaring in the garden until 4am (only stopped then when noise police intervened) when they have a party, then seem outraged when people complain. They look down on us as ours is a council house we bought. They resented having to put a fence up, even though he works in a timber yard. They do not speak to us at all. Think i will have a word with the council. Thankyou

Make sure you write a formal letter to them to show the council you have tried to communicate, it will go in your favour.
Make sure you say the boundary is their responsibility, that their rubbish will impose on your property and you are not willing to accept this. Tell them if it does fall on your property you are legally obliged to return it which you will by throwing it back into their garden. Tell them they are in breach of s8. High Hedges The Anti Social Behaviour Act 2003 by not cutting and maintianing a hedge which is on their property and has reduced the light in your garden significantly. Tell them you will give them 21 days to sort out the problems you have highlighted and if they refuse to do this you will have to contact the local authority who WILL get a court order to insist they legally cut and maintain the hedge in future. If they refuse the council will cut the hedge and they will obtain a costs order in the region of £1,800. Remain polite and factual and do not under any circumstances end up in a confrontation. Take pictures and keep a diary if necessary.

Unfortunately the intervention of the council may cost you money. It is worth talking to them first.
Best of luck, it appalls me how individuals plant these hedges then do nothing to maintain them leaving other people upset and annoyed and facing stress, a law was passed because of Leylandii, i think they should have planning permission granted with conditions of maintainance when people plant them as a hedge!