I remember having a panic about that the day before exchange on my first sale/purchase transaction.
Sold for £180k and buyers had a 10% deposit. Buying for £240k and a 10% deposit in the contract, meaning an extra £6k in deposit, which was effectively coming from the additional equity I'd built up in the first property.
Suddenly had a panic that I was going to have to find £6k to top up the difference. I'd budgeted for everything and it was a very tight move and I didn't have a penny to spare. Thankfully the solicitor explained as you have, so I was very relieved!
Solicitors would do well to have a "cheat sheet" type document that they send to all buyers/sellers of property to explain the process in a sort of idiots guide way!
@Charliechoco I know of someone recently who went from listing their property to completing on it within 7 weeks, so fast transactions can be possible, but realistically most take at least 12 weeks to get to completion and longer if there is a long chain.
There are lots of hoops that the conveyancers have to go through to make sure the property doesn't have any issues (both physically and legally), then it's a case of coordinating everything - completion dates for all the properties in the chain are the same day, mortgage offers, surveys, searches, etc. It all takes time.
If you're putting an offer in now, the conversation about dates will take place once the legal matters are underway. Have the owners of the property you want found somewhere else, for example? If they haven't they'd either have to move out in to rented or with family, or, as is more often the case, you have to wait for them to find somewhere.
Good luck OP
This is a helpful article on buying a property - Money Saving Expert