It's doable IF everyone is organised, does what they are asked to do straightaway, and keeps the pressure up on their legal team.
So solicitors need to be instructed, their instruction letter signed and and returned, and any money due up front needs to be paid before searches will be commissioned.
Some conveyancers ( not unreasonably ) do nothing until the contract package has been received from the seller, so the sellers in each part of the transaction need to make sure that the contract and supporting documents are sent out quickly, giving the buyers solicitors plenty of time to raise enquiries, obtain responses, consider them, report to their client, and possibly raise more queries after having spoken to their clients.
The buyers need to show commitment by putting their money where their mouth is, eg paying any search fees, organising proof of ID, sorting survey and mortgage.
I think it's helpful if all parties have dates to aim for - eg exchange by last week of May, completing 2 weeks after that.
Actually that's really tight, you'll need to keep pushing right from the off, and have a detailed timeline - ask if solicitors have been instructed, ask if searches applied for, check that survey happens when it is supposed to, check that everyone will have finances ready at the right time, check that no-one is going on holiday in the middle of all this.
If you want to exchange by end of May ( and you do for a mid June completion) everything will need to be ready the week before that, as it takes some days to get final paperwork signed, all deposits in place etc, removal arrangements finalised. That actually gives you about 7 weeks from now. Let all the legal teams know that this is the date the whole chain is working towards, and get the agents onside too.
Good luck, but have a Plan B, because the chances of all this happening in your required timeframe are probably less than 50:50.