Hi OP. Here is a perspective from someone who used to be a solicitor (not conveyancing though).
-As others have pointed out, the developer does not instruct the solicitor, you do. I presume the developer is the seller, so you would DEFINITELY not have the same solicitor. However, I would expect the developer to pass on your details. Even if you called to introduce yourself, that might not have been construed as a formal instruction, although take this up with the firm. You would also have had to attend the office with proof of ID and address for money-laundering purposes. Until proof of ID is received, the firm cannot act for you.
When the solicitor receives instructions, you should receive (so you should have this) a retainer letter to confirm the instruction and set out who is dealing with your work and the cost and likely timescale. Read this carefully, although I would guess you don't have this yet, as the solicitor says she was only recently instructed.
The job of the buyer's solicitor is to investigate title, carry out all searches connected with the property, liaise with the seller's solicitors. This does take some time, but you absolutely should not complete or even exchange without all of this being done.
You would never complete without having exchanged contracts first. You would have been sent a draft and asked to send in a signed copy. Before all of this, you would have had a report with the local authority searches and any other searches such as environmental ones.
You should never give up a rental unless you have exchanged and have a firm date for completion. Otherwise you risk being homeless. Try to negotiate with your landlord to stay a bit longer, though depends on whether he or she has found new tenants.
You cannot (or should not) exchange without a mortgage offer in writing.
Well in advance of exchange, you would be asked to pay the deposit monies to your solicitor's client account. The solicitor will pay that to the seller's solicitor on exchange and the mortgage company will release the balance to complete a couple of days before completion. You definitely cannot exchange/complete without having paid this money.
Most conveyancers are not qualified solicitors but they do usually have a Legal Exec qualification, although some might be unqualified paralegals. They usually have a very heavy case-load. They will not be chasing you- you chase them. Not excusing the behaviour of yours but for the conveyancer, your particular purchase will not be at the forefront of their mind. It sounds like miscommunication, probably because you were under the impression that you were completing but did not realise what this entailed.
You can change solicitors but to be honest, you are unlikely to get a Rolls-Royce service anywhere unless you pay hefty money. For a small property, it's not worth it. Stick with this firm, keep chasing them and as long as there are no hold-ups from the seller's end, all should hopefully progress once you have the search results.