You need to find out what these two issues are. Once you know, post it on here and you should get a better indication of timescales.
Why did you agree to let your sale go independent from your purchase, if this is not really what you wanted? If you don't have to be out until the end of July, surely this should be enough time to sort out the two issues and tie in completion to the same day?
By the way, if you chase, chase and chase every day, rest assured your solicitor will struggle to actually do the work.
Most people under estimate how long it takes to do the conveyancing. There is a lot of work to be covered, reviewing legal documents, obtaining information from the Council, Lender, Management Company, the Client, raising enquiries from the documents provided, checking the responses, repeating the enquiries not satisfactorily answered. Making sure money laundering ID checks are passed, obtaining funds from clients to start work, additional work might be liaising with third parties i.e. Help to Buy Equity Loan provider, Matrimonial Lawyers (in divorce cases), Equitable Charge companies (if there are charges on the title other than the mortgage), Land Registry, dealing with Gifted Deposits and obtaining ID from the giftor, making the lender aware of the gift etc. Not to mention the daily calls from every single client and EA all wanting updates because 'their case is urgent', just like everyone else's. I could go on, but no-one actually seems to want to understand that the chain only moves as quickly as the slowest responder. While the internet has speeded up response times, it has also meant that everyone seems to think that responses should be instant. I reply to emails as quickly as I can, but as soon as I have responded, the clients send in another email with yet more questions. People think nothing of reducing the price after survey results without thinking that the mortgage offer may need to be amended, the contract and transfer need to be amended, so while it doesn't take long, it does stop me from doing something else equally as urgent. Also many clients do not understand the process very well, or think they do and try to micro manage it. We get clients who think they are doing our work but actually in most cases it's just a hindrance.