I am not aware of short term rental of either USB 3G/4G dongles in the United Kingdom, and have not seen these advertised either in domestic mobile phone shops or when passing through British airports - it should be noted though that this travel has not often been for business purposes, and I already have unlimited data and wireless tethering available within Britain and some European companies which obviously means I wouldn't have been actively seeking such offers out.
There is near market saturation of smartphone ownership in Britain (93% ownership by 18-24 year olds and 81% in the 25-34 year old age group in a survey from 2013) and as such the market is highly competitive - although not as competitive as in some other European countries such as Estonia (based on month long stay by my wife last year). Several "virtual" networks (who do not own their own network infrastructure but buy access wholesale from one of the primary networks) are competing in the United Kingdom domestic which is helping keep prices down.
Smartphone and tablet usage is extremely high and there has been a push by networks to sell separate "mi-fi" hotspots for use with tablets and laptops. These are available both with pre-pay bundles or on monthly rolling contracts.
The UK network "Three" offer a monthly rolling contract on a mi-fi portable hotspot at £29.99 GBP (approximately 42.22 Eur, 5698 Japanese Yen) up front and £15.99 GBP (22.51E / 3036Y) per month for 10GB data. It would be possible to cancel the contract giving a month's notice a week in to your visit - which should mean the total is only in the region of £50GBP. Given the price of 24 hour passes at wireless hotspots at premium locations such as airports in the United Kingdom (most train stations and many pubs, cafés and restaurants will offer free wireless access) I'd imagine you'd struggle to find wireless hotspot rental much cheaper - if indeed the option is available. The data allowance is generous, and you'd own the device (which is sold unlocked) at the end of the contract. This device isn't 4G capable (although your contract would be as Three don't charge extra for 4G data usage), but has a generous data allowance and is theoretically capable of 21.6Mbps throughput according to the manufacturer. If 4G speed is more important than data allowance then there is a 4G option with half the data allowance available at the same monthly price - and slightly more upfront for the device, which has better specifications. I'm not an expert but there appears to be some frequency overlap with Japanese frequencies so it is possible that the devices might work back home in Japan - but even if not would be worth holding on to for future trips to Europe.
The UK's largest mobile provider EE has an even more generous 15 GB allowance 4G 30 day rolling contract mobile wireless hotspot available for £20GBP per month (28.16E / 3799Y) and £39.99 GBP (56.29E / 7596Y) upfront. The device may be locked though, so be aware and ask before you buy.
If wireless isn't important to you and you only plan to use the internet with a laptop then it would be cheaper in terms of upfront cost to buy a USB modem/ dongle (it may be worth noting that these are probably the first thing most people will think of given the word "dongle") which is available in both 3G and 4G variants.
Depending on your residence arrangements whilst you're staying in the UK, it may not be possible to obtain even a thirty day contract if you don't have an address to give the network. The absence of any history of credit in the United Kingdom might be a factor, but the risk to the network is relatively low with the device pre-paid and a low monthly cost are in your favour here - you're not asking for a new iPhone or similarly expensive piece of hardware. Another limiting factor might be a possible requirement for a UK bank account to pay the bill - although it might be possible to pay using a credit and/or debit card over the phone.
As a backup, should any of these factors make it difficult to get a contract set up, you can obtain the same hardware as a pre-paid bundle costing £59.99 GBP (84.44E / 11394Y) for the 3G hardware and £69.99 GBP (98.52E / 13292Y) for the 4G. These both only come with 3GB data allowance, as it sounds like you need more you might need to bump up to the 12 month pre-paid bundle at £109.99 GBP (154.85E / 20891Y) for 3G or (168.93E / 22790Y) for 4G.
It might be worth contacting Three in advance (they have online chat or email) to see what you can do or bring with you from Japan to help obtain the thirty day contract as your data needs in the UK seem significant and the contract offers much more data at less cost. Their website does specifically mention contacting them by phone if you don't have UK address history, so there is reason for hope.