The good news is it's doubtful you have anything at all to worry about in relation to the business names. The bad news is you've been going about things the wrong way by altering the names by one letter. The neutral news is that it doesn't really matter that you've done that; all you have done is inconvenience yourself unnecessarily.
The names of your two businesses are completely irrelevant, and your choice to differ the names by one letter is of no consequence (in this context at least; issues could arise under passing off / trademark laws but that isn't within scope here). For the vast majority of limited companies, the name must contain the word "Ltd" or "Limited" at the end and this alone would distinguish your two names even without a letter being different. Home Solution, Home Solutions, Home Solution Limited, and Home Solutions Limited are four different business names. There is an exception to this rule in Section 60 of the Companies Act 2006 but you would know if this applies to your case because you would have declared it to Companies House (and chosen a name without Ltd / Limited at the end).
What matters for trading and tax purposes is which entity is carrying on the trade in reality. In the case of the sole proprietorship, the entity is you as a natural person. In the case of the limited company (which has its own separate legal personality), it is the company.
This will be determined according to which entity contracts with third parties during the course of trade i.e. which entity promises to provide goods/services to customers, invoices and receives payment from customers, receives promises for provision of goods/services from suppliers, and receives invoices and makes payments to suppliers.
If you are doing things in accordance with the law, it will usually be abundantly clear which entity is doing the trading. That's because the law requires you to provide information about your entity during the course of trading in sufficient detail that it is possible to identity the correct one. The exception would be if your business consists solely of oral contracts and you never follow those contracts up with any written correspondence or invoices (which is in any case highly inadvisable for most types of businesses).
In the case of a limited company, the company is required to state its registered name (which, remember, includes the word "Ltd" or "Limited") on virtually all forms of written communications (including letters and invoices). In addition, on all business letters, order forms, and websites, the company is required to state (among other items) its registered number and registered office address. See Part 6 of the Company, Limited Liability Partnership and Business (Names and Trading Disclosures) Regulations 2015, and in particular Regulations 24 and 25.
In the case of a sole proprietorship, the person running the business is required to state their individual name and an address for service in the UK (for a UK based business) on all business letters, orders for goods and services, invoices, receipts, and written demands for payment. See Chapter 2 of Part 41 of the Companies Act 2006, and in particular Sections 1201 and 1202.
If HMRC were ever to look into your affairs, they would simply check your business documents and contracts to identify which entity was doing the trading in reality. The most obvious first place to look would be the contract with your bank: which entity opened the bank account which is being used to conduct the trade? The second obvious places are correspondence with customers and suppliers.
The only time you will have a problem is if you've failed to comply with the disclosure requirements in the applicable Act / Regulations, and it's therefore not clear which entity was trading. For example, if you simply write "Home Solutions" (with none of the other required information) on your invoices then it is potentially debatable whether that was meant to be Home Solution (with the "s" added in error) or Home Solutions Limited (with Limited omitted in error).
Bottom line: keep your paperwork in order and make sure that the correct entity is named everywhere (paying particular attention to your contracts as these identify conclusively the trading entity). Have a standard wording for this that you keep on a letterhead, template, email signature, text file for copy/pasting, etc. A typical wording for a sole proprietorship is "Joe Bloggs of 15 The Avenue, London, N1 2AB, trading as Home Solution". Open up a sole proprietorship bank account and use it exclusively for the business. Avoid mixing in transactions belonging to a different entity or personal transactions, and avoid trading via a different entity's bank account. The latter can be fine (e.g. agency relationships) but you need to know what you are doing.
Don't worry about how different the names are; certainly don't try to "hack" a solution to an XY problem by altering one letter in the names unless you have other reasons for doing so.