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Currently, BTC has a problem with small transfers, i.e sending £1 will cost £4 which is just ridiculous. Will XMR ever have the same problem? I know LTC have sloved this issue and Bitcoin cash has also solved this issue?

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Assuming the block sizes and average block times are given, a cryptocurrency has a well defined maximum average throughput byte-wise. It's true for Bitcoin (and its forks, of course). See the edit for Monero block sizes - they aren't limited, but the miners are discouraged from increasing them too fast. Assuming all the transactions are of roughly same size, there is a maximum throughput in transactions per second. It gets higher if blocks are bigger, or transactions are smaller, or blocks are more frequent.
Now suppose we hit this maximum in terms of announced transactions per second, and miners can't fit all the announced transactions into blocks. The rational behavior for miner, given a constant block size, is to include the transactions with highest fee per byte (that's why they don't care that you "only" try to send £1 - this transaction is as big in terms of bytes as sending £1,000,000). In Bitcoin, the maximum theoretical average is ~10 tx/s, and on practice it's even lower as many transactions aren't of a minimum size. Once the currency got popular enough, the demand arguably got bigger than that. Note that some people claim that the transaction pool was spammed during the discussion on bigger blocks, exactly to make a point that this problem is real - whether or not it is true, I don't know.
Bitcoin Cash aims to solve this problem by having bigger blocks. Essentially, they solved it by not having that high of a demand for transactions. Litecoin has much higher blocks frequency, as well as lower demand for transactions.
Monero is in a good place in terms of fees per byte, but the bytes per transaction ratio is relatively high (cost of anonymity). It is actively worked on - with Bulletproofs, the transactions will get much smaller. It may not change fees per byte, but it greatly reduces fees per transaction - which is relevant in scope of your question.
Monero can get the same problem as the one recently encountered by Bitcoin, if the demand for transactions suddenly goes far beyond the throughput of the system. It doesn't seem to be the case as of now.

Edit: This is a simplified explanation without considering Monero dynamic block size algorithm. The described trade-offs seemingly still apply, so leaving it for now.

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    This answer doesn't describe the dynamic block size algorithm of Monero and is therefore, in my opinion, inaccurate. See: getmonero.org/2017/12/11/A-note-on-fees.html – dEBRUYNE Jan 2 '18 at 13:52
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    @dEBRUYNE You're totally right, thanks, forgot about it. Please suggest an edit if applicable, my understanding of the influence of the algorithm on the model described above is probably too limited. – Valgrind Jan 3 '18 at 0:40

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