Building on the great answer by Andrew:

1) What is this "neutrino" refered above?

Where as BIP 37 generates a client side Bloom filter including the HD wallet addresses, BIP 157/158 "Neutrino" creates a server side [can be any Bitcoin full node] Golomb-Rice filter including all transactions in a valid Bitcoin block. You are no longer querying if a specific transaction is in a block, which might reveal which transactions you are interested in, but rather you query which transactions are in a block, and then you download the interesting blocks from another node. Every valid block has only one correct BIP 158 filter, and you can thus cross reference against several different nodes.

2) When is it expected to fully work , on what is timeline dependent?

I don't know how fast the core process will integrate BIP 157/8 [probably in two weeks?!], but because this is not a consensus critical change, it can be implemented by individuals much earlier. For example, Wasabi Wallet is creating Golomb-Rice filters on the centralized Wasabi back-end server, the individual wallets are checking which block contains their interested transactions, then the wallet is downloading that block on the Bitcoin P2P network over tor, or from your own full node. This provides great privacy improvements, as neither the Wasabi back-end nor the P2P network know exactly which blocks [due to tor / your own node] or addresses [due to BIP 157/8] you are interested in.

Further, Lightning Labs has a testnet version of their app which is running Neutrino, their implementation of BIP 157/8.

3) And to what extent/how would this "neutrino" change the "Game" as implied above? What effects would it have on the way lightning works?

As Andrew points out, running your own full node and verifying the entire block chain is the best practice for both security and privacy. With SPV proofs you inherently "trust the miners" that the chain with most accumulated PoW is valid. "Inverted" block filters protect wallet privacy and is a great improvement to address filters. I'd say that in all regards: Own full node >>> Golomb-Rice filters > Bloom filters.

1) What is this "neutrino" refered above?

Neutrino is an addition to the Bitcoin P2P network protocol which enables more private SPV wallets. How it works is that nodes will create a "filter" for each block and relay those filters to SPV wallets. Those SPV wallets can then check if any of their addresses appear in the filter. If they do, they can request the entire block and pull the transactions that they want from that block. This protocol is described in BIP 157 and the filter construction in BIP 158.

2) When is it expected to fully work , on what is timeline dependent?

It is dependent on support being merged into full node software. This is a network protocol feature and must be supported by full nodes. Currently the predominant full nodes software in use on the Bitcoin network is Bitcoin Core. There is active and ongoing work to get BIP 157/158 into Bitcoin Core. There are other software that support BIP 157/158 but they are not widely used. There is no current roadmap or timeline for BIP 157/158 inclusion in Bitcoin Core.

3) And to what extent/how would this "neutrino" change the "Game" as implied above? What effects would it have on the way lightning works?

Neutrino would make it possible to run Lightning Network software on lower powered devices such as mobile phones. It makes it possible to not need the Bitcoin blockchain or to not receive every single block and transaction yet still be fairly certain of the security of your Bitcoin and lightning channels. Of course it is not as secure as running your own full node, but for devices where this is not possible, BIP 157/158 would make it possible to run your own lightning wallet without fully trusting a third party.